System for analyzing vascular refill during short-pulse ultrafiltration in hemodialysis

ABSTRACT

A method includes: receiving measurements of a blood-related parameter corresponding to a patient undergoing hemodialysis; estimating a value of one or more hemodialysis treatment-related parameters by applying a vascular refill model based on the received measurements of the blood-related parameter, wherein the one or more hemodialysis treatment-related parameters are indicative of an effect of vascular refill on the patient caused by the hemodialysis; determining, based on the one or more estimated values of the one or more hemodialysis treatment-related parameters, a hemodialysis treatment-related operation; and facilitating performance of the treatment-related operation. The vascular refill model is a two-compartment model based on a first compartment corresponding to blood plasma in the patient&#39;s body, a second compartment based on interstitial fluid in the patient&#39;s body, and a semi-permeable membrane separating the first compartment and the second compartment.

CROSS-REFERENCE TO RELATED APPLICATIONS

This patent application is a continuation of U.S. patent applicationSer. No. 15/309,727, filed Nov. 8, 2016, which is a national phase ofInternational Patent Application No. PCT/US2015/033225, filed May 29,2015, which claims the benefit of U.S. Provisional Patent ApplicationNo. 62/005,744, filed May 30, 2014. All of the aforementioned patentapplications are incorporated herein by reference in their entireties.

BACKGROUND

End-stage renal disease (ESRD) patients typically have an increasedextracellular volume (ECV) due to their impaired kidney function.Management of this fluid excess is one of the cornerstones in thetreatment of these patients. In patients who undergo hemodialysis (HD),this excess extracellular fluid volume can be removed by ultrafiltration(UF). During UF, fluid is removed from the blood stream (intravascularcompartment), and fluid from the tissue (interstitial compartment)shifts into the intravascular space (driven by hydrostatic and oncoticpressure gradients; details below) to counter the reduction in bloodplasma volume. This process, called vascular refilling, is critical formaintenance of adequate intravascular filling and blood pressure duringdialysis.

Whenever the vascular refill rate is less than the ultrafiltration rate,the plasma volume declines; this process manifests itself in a declinein absolute blood volume (ABV) and a decline in relative blood volume(RBV). This decline of RBV translates into increased hematocrit andblood protein levels. Measurements of hematocrit or blood proteinconcentration during HD form the basis of relative blood volumemonitoring. RBV can be measured continuously and non-invasivelythroughout HD with commercially available devices, such as the Crit-LineMonitor (CLM) or the Blood Volume Monitor (BVM). While the CLM measureshematocrit, the BVM measures blood protein concentration.

The RBV dynamic is the result of plasma volume reduction byultrafiltration, and vascular refilling by capillary and lymphatic flow.

SUMMARY

Embodiments of the invention provide a system for analyzing refillprocesses in patients. Understanding these quantitative aspects isclinically important, since both the driving forces (e.g. hydrostaticpressures; details below) and the capillary tissue characteristics (e.g.hydraulic conductivity; details below), are intimately related to(patho) physiological aspects which are highly relevant in the care ofHD patients, such as fluid overload and inflammation. Neither of theseforces and tissue characteristics are accessible to direct measurementsfeasible during routine HD treatments.

The system utilizes mathematical models on qualitative and quantitativebehavior of vascular refill during dialysis to estimate certain outputparameters corresponding to the quantities that are indicative of thefluid dynamics within a patient. Based on these output parameters, thesystem is able to perform various treatment-related operations, such asindicating status of the parameters to a treating physician, providingnotifications and alerts, adjusting current and/or future treatmentprocesses, aggregating and storing patient-specific information toprovide trend data and/or to modify future treatments based thereon,etc.

In a particular exemplary embodiment, the system utilizes atwo-compartment model incorporating microvascular fluid shifts and lymphflow from the interstitial to the vascular compartment. Protein flux isdescribed by a combination of both convection and diffusion.

In an exemplary embodiment, a Crit-Line device is used to identify andmonitor certain input parameters of a patient, including for example, aHematocrit (“Hct”) level.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention will be described in even greater detail belowbased on the exemplary figures. The invention is not limited to theexemplary embodiments. All features described and/or illustrated hereincan be used alone or combined in different combinations in embodimentsof the invention. The features and advantages of various embodiments ofthe present invention will become apparent by reading the followingdetailed description with reference to the attached drawings whichillustrate the following:

FIG. 1 is a block diagram illustrating an exemplary network environmentusable in connection with certain exemplary embodiments of theinvention.

FIG. 2 is a flowchart illustrating an exemplary process for obtaininginput parameters.

FIG. 3 is a flowchart illustrating an exemplary process for a server toperform computations based on a vascular refill model.

FIG. 4 is a flowchart illustrating an exemplary process for utilizingthe output parameters computed by the server.

FIGS. 5 and 6 illustrate an exemplary reporting interface for certainoutput parameters.

FIG. 7 is a model diagram depicting the fluid movement in thecompartments.

FIG. 8 illustrates the dynamical behavior of the model state variablesfor an hour where J_(UF)=30 mL/min for 20 minutes on t∈[20, 40] andJ_(UF)=0 for t∈[0, 20) and t∈(40,60].

FIG. 9 illustrates hematocrit levels as model output during a rest phasefor t∈[0, 20) minutes, UF at J_(UF)=30 mL/min for t∈[20, 40] minutes,and refill phase for t∈(40, 60] minutes.

FIGS. 10 and 11 are plots illustrating traditional sensitivities ofmodel output with respect to certain parameters.

FIG. 12 is a graph illustrating an exemplary model output where themodel is adapted to the hematocrit measurements of a specific patient(black curve) by identifying L_(p) and P_(c). The parameters wereestimated within the two vertical dashed lines and hematocrit valueswere predicted for the following 20 minutes (white curve).

FIG. 13 is a graph illustrating an exemplary model output where themodel is adapted to the hematocrit measurements of another specificpatient (black curve) by identifying L_(p) and P_(c), and κ. Theparameters were estimated within the two vertical dashed lines andhematocrit values were predicted for the following 20 minutes (whitecurve).

FIG. 14 illustrates plots of the functions π_(p) and π_(i) referenced inAppendix A.

FIG. 15 illustrates graphs of π_(p), π_(p,approx), π_(i) andπ_(i,approx) referenced in Appendix A.

FIG. 16 illustrates errors for the quadratic approximation of π_(p) andπ_(i) referenced in Appendix A.

DETAILED DESCRIPTION

FIG. 1 is a block diagram illustrating an exemplary network environmentusable in connection with certain exemplary embodiments of theinvention. The system includes a patient monitoring system 101 (forexample, a combination of a sensing device connected to a host computerhaving a display for indicating patient-related or treatment-relatedinformation and having a network communication interface, or aintegrated sensing device with communication capabilities), typicallylocated at a dialysis treatment center 102, that is configured totransmit hematocrit (Hct) (or alternatively RBV), ultrafiltration rate(UFR), and patient identification (pID) information over a network 110.Examples of patient monitoring systems usable with embodiments of theinvention include, but are not limited to, a Crit-Line monitoringdevice, a CliC monitoring device, and other devices suitable formeasuring Hct and/or RBV. A server 120 receives, via the network 110(e.g., the internet or a local or private network), the Hct (oralternatively RBV) and UFR values. The server may also utilizepatient-specific data retrieved from a data warehouse 130 (e.g., adatabase in communication with the server 120) based on the pID. Thepatient-specific data may include, for example, ABV, bioimpedancemeasurements, height, weight, gender, IDWG, as well as previousestimated values for L_(p), P_(c), P_(i), σ, α, κ (as discussed below)determined for the patient.

The server 120 uses the received information to calculate furtherinformation based upon models for vascular refill during dialysis (e.g.,estimated values for L_(p), P_(c), P_(i), σ, α, κ and trend data). Thisinformation may then be provided to the data warehouse 130 for storageand for future reference, and to the dialysis center 102 for indicationto a treating physician or for performance of other treatment-relatedoperations (e.g., providing notifications and alerts, and adjustingcurrent and/or future treatment processes).

Although FIG. 1 depicts a network environment having a server 120 anddata warehouse 130 remotely situated from the dialysis center 102, itwill be appreciated that various other configurations of the environmentmay be used as well. For example, the computing device performing themodel-based estimations may include a local memory capable of storingpatient-specific data, and the computing device may be situated locallywithin the dialysis center and/or formed integrally with the patientmonitoring system (e.g., as part of a host computer or integratedsensing device). In another example, the patient-specific data may bestored on a data card or other portable memory device that is configuredto interface with a treatment device, to allow the treatment device toprovide patient-specific treatment and display patient-specificinformation.

FIG. 2 is a flowchart illustrating an exemplary process for obtaininginput parameters. The input parameters, for example, may be received bythe server 120 and then used in the model-based estimations performed bythe server 120. At stage 201, a patient visits the dialysis center 102,and at stage 203, a dialysis treatment for the patient is started. Atstage 205, a patient ID corresponding to the patient is sent to the datawarehouse 130, and at stage 207, certain patient data is communicated tothe server 120. The patient data that may be passed on includes—ifavailable for the patient—patient ID, absolute blood volume (ABV) andbioimpedance data, gender, weight, height, intradialytic weight gain(IDWG) and previous values for the indicators L_(p), P_(c), P_(i), σ, α,κ.

Additionally, in the meantime, a sensing device collects data forhematocrit (Hct) and/or relative blood volume (RBV) at stage 209. Thisdevice can be, for instance a Crit-Line Monitor or one of its successors(e.g. CliC device) or any other machine that measures either Hct or RBVwith sufficient accuracy and frequency (e.g., at least 1 measurement perminute). The more accurate the measurement is, the more parameters canbe identified and estimated by the server 120. After a predefined timehas passed (e.g., between 20 to 50 minutes), at stage 211, the collectedHct or RBV together with the ultrafiltration profile (including the UFR)that was run up to that point and the patient ID is sent to the server120. The server then uses the data corresponding to the treatment of thepatient, as well as patient data from the data warehouse 130 (ifavailable), to perform model-based computations (discussed in moredetail below with respect to FIG. 3).

It will be appreciated that the patient ID may be used by the server 120to merge the data from the clinic (the dialysis treatment center 102)with the patient information obtained from the data warehouse 130.

FIG. 3 is a flowchart illustrating an exemplary process for a server toperform computations based on a vascular refill model. The computationsinclude processing the received information and computing estimates forthe indicators L_(p), P_(c), P_(i), σ, α, and/or κ. If previousparameter estimates exist for the patient (e.g., L_(p), P_(i), P_(i), σ,α, and/or κ values for the patient received from the data warehouse 130)at stage 301, the server at stage 303 may use those previous parametersas a starting point. If previous parameter estimates do not exist forthe patient at stage 301 (e.g., for a new patient), default initialparameters may be used as the starting point (see Table 1 below) atstage 305.

Using the determined starting point, the server 120 then utilizes amathematical model for vascular refill (as will be discussed in furtherdetail below) to estimate values for output parameters (or“hemodialysis-related treatment parameters”) L_(p), P_(c), P_(i), σ, α,and/or κ (which are indicative of an effect of vascular refill on thepatient caused by the hemodialysis). This includes performing aparameter identification on the desired time interval at stage 307,solving model equations using the initial parameter values at stage 309,plotting the model output with Hct data at stage 311, and determiningwhether the model fits the data at stage 313. If the model does not fitthe data at stage 313, the initial values are modified at stage 315 andstages 307, 309, 311 and 313 are performed again. If the model does fitthe data at stage 313, but checking the range of parameter valuesobtained to determine whether the values are within a(patho)physiological range at stage 317 reveals that the values are notwithin range the (patho)physiological range, the initial values aremodified at stage 315 and stages 307, 309, 311 and 313 are performedagain. If the model does fit the data at stage 313, and checking therange of parameter values obtained to determine whether the values arewithin a (patho)physiological range at stage 317 reveals that the valuesare within the (patho)physiological range, the server 120 provides oneor more estimated output parameters as output so as to facilitate theperformance of one or more treatment-related operations (as will bediscussed below in further detail with respect to FIG. 4).

The parameter identification at stage 307 involves an inverse problembeing solved several times (as will be discussed in further detailbelow). Additionally, solving the model equations at stage 309 involvessolving the inverse problem to compute parameter estimates for L_(p),P_(c), P_(i), σ, α, and/or κ. Based on checking whether the model fitsthe data at stage 311, as well as checking whether the values are withina (patho)physiological range at stage 317, the computation process isrepeated until reliable and meaningful parameter values are found.

FIG. 4 is a flowchart illustrating an exemplary process for utilizingthe output parameters computed by the server. In an exemplaryembodiment, the server is able to estimate the following indicatorswithin an hour of beginning hemodialysis: L_(p), P_(c), P_(i), σ, α,and/or κ. At stage 401, the estimated output parameters are output viathe network 110 and communicated to the dialysis center 102. Theestimated output parameters may also be output via the network 110 tothe data warehouse 130 and stored. Trend data based on the estimatedoutput parameters in combination with previously estimated outputparameters for the same patient may also be output and stored at stage403. Using the estimated output parameters from stage 401 and/or thetrend data at stage 405, a current hemodialysis treatment and/or futurehemodialysis treatment may be modified (for example, manual adjustmentsto ultrafiltration rate and/or treatment time made based on a treatingphysician's review of the data, and/or automatic adjustments made basedon the output parameters meeting certain criteria such as exceedingcertain thresholds or falling outside of certain ranges). In oneexample, treatment may be automatically stopped or slowed if theestimated values indicate that continued treatment at a current UFR isdangerous to the patient.

Additionally, notifications and/or alerts may be generated at stage 407.For example, treating physicians and other personnel may be notified ofthe estimated output parameters based on the display of the values forthe estimated output parameters on a screen of a computing device at atreatment facility. They may also be otherwise notified via variousforms of messaging or alerts, such as text messaging, paging, internetmessaging, etc. Specific alerts, for example, relating to potentialproblems arising from the hemodialysis treatment, may be generated basedon certain output parameters meeting certain criteria (such as exceedingor falling below a predetermined range of values or threshold, orrising/falling at a rate determined to be potentially problematic). Inan example, the values of certain output parameters are presented on ascreen (e.g. the display of the Crit-Line Monitor) together with normalranges. Further, trends of the parameter values for the patient (e.g.,over the last 1-3 months) may also be depicted. Examples of graphicaldepictions on such a display are illustrated in FIGS. 5 and 6 (whichillustrate an exemplary reporting interface for certain outputparameters).

When there exists previous parameter estimates for L_(p), P_(c), P_(i),σ, α, κ for the patient, the trend over a time period (e.g., the last1-3 months) can be computed using linear regression. The new parametervalues (together with the trend, if available) are passed on to bereported at the clinic. Moreover, the new estimates are communicated tothe data warehouse and stored, such that the information about theindicators will be made accessible for additional analyses. Theseadditional analyses include but are not limited to trend analysis overtime, correlational analysis with other variables, such as interdialyticweight gain, target weight, and biomarkers, such as serum albuminlevels, neutrophil-to-lymphocyte ratio, C-reactive protein (CRP), andothers.

Since the identified variables are indicative of (patho)physiologicalprocesses, but are not accessible to direct measurements, the estimatedvalues will be considered for clinical decision making. For example, ahigh value for the filtration coefficient (L_(p)) is indicative ofinflammation, which may require additional investigation to confirm thepresence of inflammation. Trend data showing rising levels of L_(p) mayfurther be indicative of smoldering or aggravating inflammation and mayrequire additional investigation as well. Thus, certain notifications oralerts/alarms may be triggered based on the value for L_(p) exceeding apredetermined threshold or the rate of increase for L_(p) exceeding apredetermined threshold.

In another example, a low value for the systemic capillary reflectioncoefficient (σ) is indicative of capillary leakage, sepsis, or anallergic response and/or anaphylaxis, which may require additionalinvestigation into the source of the leakage, sepsis, or allergicresponse. Trend data showing falling levels of (σ) is indicative ofsmoldering or aggravating capillary leakage. Thus, certain notificationsor alerts/alarms may be triggered based on the value for a being below apredetermined threshold or the rate of decrease for a falling below apredetermined threshold.

In another example, a high value of (or increasing trend for)hydrostatic capillary pressure (P_(c)) is indicative of autonomicdysfunction, high venous pressure, drugs, or arterial hypertension,which may require evaluation of a patient's drug prescription and/or acardiac exam to investigate the high venous pressure. On the other hand,a low value of P_(c) is indicative of an exhausted reserve to increaseperipheral resistance, which may require measures to increaseintravascular volume (e.g., lowering the UFR). Thus, certainnotifications or alerts/alarms may be triggered based on the value forP_(c) being outside a predetermined range or the rate of increase forP_(c) exceeding a predetermined threshold. Treatment adjustments mayalso be made based on P_(c) falling below a predetermined threshold,such as automatically decreasing the UFR for a current or a futuretreatment of the patient. The UFR may also be manually decreased by atreating physician, for example, in response to reviewing the P_(c)information displayed at the treatment center, or in response to anautomatic prompt triggered by the detection of the low P_(c) level thatgives the physician the option of decreasing the rate of and/or stoppingtreatment.

In yet another example, a high value of (or increasing trend for)hydrostatic interstitial pressure (P_(i)) and/or constant lymph flowrate (κ) is indicative of interstitial fluid overload, while a low value(or decreasing trend for) hydrostatic interstitial pressure (P_(i))and/or constant lymph flow rate (κ) is indicative of interstitial fluiddepletion. The clinical response here may be to re-evaluate the fluidremoval rate (i.e., increasing it in the event of fluid overload anddecreasing it in the event of fluid depletion) for a current and/orfuture treatment. As discussed above with respect to P_(c),notifications and/or alerts/alarms may be triggered based on thehydrostatic interstitial pressure (P_(i)) and/or constant lymph flowrate (κ) falling outside respective predetermined ranges, and automaticor manual treatment modifications may be made as well.

It will be appreciated that the predetermined thresholds or ranges usedin the aforementioned comparisons may be based on previous patient data,such that a predetermined threshold for one patient may differ from thepredetermined threshold for another patient. Thus, outlier values withrespect to the estimated output parameters may be detected and respondedto appropriately (e.g., with a notification or alert/alarm, or withadjustment of a current and/or future treatment).

As discussed above, significant changes of these variables betweendialysis sessions, marked trends or out of range values may behighlighted by a device at the treatment center pursuant to stage 407.In one example, an alarm flag can be used to mark questionableparameters needing further investigation by clinic personnel (e.g., byalerting clinic personnel through visual and/or audio alarms triggeredby the patient monitoring device). For instance, as discussed above, apositive P_(i) may indicate fluid overload and may be considered whentarget weight is prescribed. Another example is an increase of L_(p),which may indicate an evolving inflammatory process. Such a signal mayresult in additional diagnostic interventions, such as measurement ofCRP, clinical evaluation, blood cultures, or medical imaging.

Additionally, the output parameters discussed herein (L_(p), P_(c),P_(i), σ, α, and/or κ) may further serve as independent variables instatistical models designed to predict patient outcomes of interest. Forexample, the server of FIG. 1 or a separate external computing devicemay access the data warehouse to obtain stored parameters pertaining toa patient and make predictions regarding corresponding characteristicsand trends pertaining to that patient based thereon. An illustration ofrelevant prediction models is provided by the discussion of logisticregression models presented in Thijssen S., Usvyat L., Kotanko P.,“Prediction of mortality in the first two years of hemodialysis: Resultsfrom a validation study”, Blood Purif 33:165-170 (2012), the entirety ofwhich is incorporated by reference herein. The predictors used in thosemodels were age, gender, race, ethnicity, vascular access type, diabeticstatus, pre-HD systolic blood pressure, pre-HD diastolic blood pressure,pre-HD weight, pre-HD temperature, relative interdialytic weight gain(of post-HD weight), serum albumin, urea reduction ratio, hemoglobin,serum phosphorus, serum creatinine, serum sodium, equilibratednormalized protein catabolic rate, and equilibrated dialytic and renalK*t/V (K being the combined dialytic and renal clearance for urea, tbeing treatment time, and V being the total urea distribution volume).Analogously, the parameters discussed herein (e.g., L_(p), P_(c), P_(i),σ, α, κ) may be used as predictors in such models, either by themselvesor alongside other predictors, similar to other predictors used inpredictive statistical models.

It will be appreciated that the referenced logistic regression modelsare only exemplary. Various different types of statistical models may beused, with categorical or continuous outcomes. Examples of modelsinclude Cox regression models, Poisson regression models, acceleratedfailure time models, generalized linear models, generalized additivemodels, classification trees, and random forests. Examples of outcomesof interest include death during a certain period of time,hospitalization (binary or count) over a certain period of time,systemic inflammation (as measured by biochemical markers, such asC-reactive protein and IL-6), and degree of fluid overload (asdetermined by bioimpedance or other methods).

Principles underlying the operation of the server depicted in FIG. 1, aswell examples verifying these principles, are discussed in the followingdisclosure and in the Appendices.

Modeling Assumptions and Formulation:

The model of vascular refill presented herein is a two-compartmentmodel. The blood plasma in the patient's body is lumped in onecompartment and the interstitial fluid including the lymphatic systemare lumped in another, namely, the plasma and interstitial compartments,respectively. The plasma and interstitium are separated by a capillarywall which is a semipermeable membrane regulating fluid exchange andprotein flux. Fluid movement between plasma and interstitium isinfluenced by the properties of the capillary wall (reflectioncoefficient σ and filtration coefficient L_(p)), and pressure gradientsacross the membrane (oncotic and hydrostatic pressures). Furthermore,lymph flows at a constant rate and protein concentration from theinterstitial into the plasma compartment. The model is formulated todescribe the short-term dynamics of vascular refill for a period ofabout one hour. Hence, some of the model assumptions are only valid whenconsidering a short time duration. FIG. 7 is a model diagram depictingthe fluid movement in the compartments.

Assumptions:

(1) The plasma compartment (V_(p)) is connected to the interstitialcompartment (V₁) which includes the lymphatic system in themicrovasculature. V_(p) is open at the dialyzer membrane whereprotein-free ultrafiltrate is removed during ultrafiltration.(2) The ultra filtration rate (J_(UF)) set at the dialysis machinedetermines the flow across the dialyzer.(3) A constant lymph flow (κ) with constant protein concentration (α)goes from interstitium into plasma.(4) Net flow between V_(p) and V_(i) is determined by the Starlingpressures.(5) Colloid osmotic pressure relationships are determined by proteinconcentrations.(6) The hydrostatic pressure gradient is assumed to be constant.(7) The hydrostatic capillary pressure (P_(c)) is constant.(8) The net protein flux (J_(s)) is the sum of both convective anddiffusive fluxes across the capillary wall.

By Assumptions 1-3, the change in plasma volume at time t is governed by

$\begin{matrix}{{\frac{{dV}_{p}(t)}{dt} = {{J_{v}(t)} + \kappa - {J_{UF}(t)}}},} & (1)\end{matrix}$

where J_(v)(t) represents the amount of fluid crossing the capillarymembrane at a certain time t, K is the lymph flow from interstitium toplasma, and J_(UF)(t) is the ultrafiltration rate. The fluid movementacross the membrane depends on the net imbalance between effectivecolloid osmotic and pressure gradients (Assumption 4). FollowingStarling's hypothesis, we have

J _(v)(t)=L _(p)(σ(π_(p)(t)−π_(i)(t))−(P _(c)(t)−P _(i)(t))).  (2)

with L_(p) denoting the filtration coefficient (which is hydraulicconductivity×surface area), a is the osmotic reflection coefficient,π_(p)(t), π_(i)(t) are the plasma and interstitial colloid osmoticpressures, respectively, and P_(c)(t), P_(i)(t) are the hydrostaticcapillary and interstitial pressures, respectively, at a given time t.Plasma proteins leak into the interstitium and the degree of leakinesscan be quantified by Staverman's osmotic reflection coefficient σranging from 0 to 1; where a value σ=1 means perfect reflection, andthus no leakage of the specified solute. A quadratic polynomialapproximation is used to describe oncotic pressures, though otherapproximations are possible:

π_(p)(t)=α_(p) ₁ c _(p)(t)+α_(p) ₂ c _(p)(t)²,

π_(i)(t)=α_(i) ₁ c _(i)(t)+α_(i) ₂ c _(i)(t)².  (3)

where c_(p)(t), c_(i)(t) are protein concentrations in plasma andinterstitium, respectively, at a given time t. Further details can befound in Appendix A.

To describe capillary refill dynamics during short-pulseultrafiltration, it is assumed that the pressure difference betweenP_(c)(t) and P_(i)(t) is constant (Assumption 6). Since P_(c) is wellautoregulated over a wide range of blood pressures, it is furtherassumed that it remains constant for short duration, that is,P_(c)(t)≈P_(c) (Assumption 7). As a consequence of Assumptions 6 and 7,P_(i)(t) is constant during short-time duration, that is,P_(i)(t)≈P_(i).

The net flux of proteins between plasma and interstitium is the sum ofconvective and diffusive fluxes across the capillary wall and proteinbackflow from the lymph (Assumption 8). Thus, we have

$\begin{matrix}{{J_{s}(t)} = \left\{ \begin{matrix}{{{J_{v}(t)}\left( {1 - \sigma} \right){c_{i}(t)}} - {{{PS}\left( {{c_{p}(t)} - {c_{i}(t)}} \right)}\frac{x(t)}{e^{x{(t)}} - 1}} + {\alpha \; \kappa}} & {{{{if}\mspace{14mu} {J_{v}(t)}} > 0},} \\{\alpha\kappa} & {{{{if}\mspace{14mu} {J_{v}(t)}} = 0},} \\{{{J_{v}(t)}\left( {1 - \sigma} \right){c_{p}(t)}} - {{{PS}\left( {{c_{p}(t)} - {c_{i}(t)}} \right)}\frac{x(t)}{e^{x{(t)}} - 1}} + {\alpha \; \kappa}} & {{{{if}\mspace{14mu} {J_{v}(t)}} < 0},}\end{matrix} \right.} & (4)\end{matrix}$

where PS is the permeability-surface area product, a is theconcentration of protein backflow from the lymph, and x is the Pecletnumber describing the convective flux relative to the diffusive capacityof the membrane:

$\begin{matrix}{{x(t)} = {\frac{{J_{v}(t)}\left( {1 - \sigma} \right)}{PS}.}} & (5)\end{matrix}$

When J_(v)(t)≥0, protein flows into the plasma while when J_(v)(t)<0protein goes into the interstitium. Equation (4) can be rewritten as

$\begin{matrix}{{J_{s}(t)} = \left\{ \begin{matrix}{{{J_{v}(t)}\left( {1 - \sigma} \right)\left( {{c_{i}(t)} - \frac{{c_{p}(t)} - {c_{i}(t)}}{e^{x{(t)}} - 1}} \right)} + {\alpha \; \kappa}} & {{{{if}\mspace{14mu} {J_{v}(t)}} > 0},} \\{\alpha\kappa} & {{{{if}\mspace{14mu} {J_{v}(t)}} = 0},} \\{{{J_{v}(t)}\left( {1 - \sigma} \right)\left( {{c_{p}(t)} - \frac{{c_{p}(t)} - {c_{i}(t)}}{e^{x{(t)}} - 1}} \right)} + {\alpha \; \kappa}} & {{{if}\mspace{14mu} {J_{v}(t)}} < 0.}\end{matrix} \right.} & (6)\end{matrix}$

Note that J_(s)(t) is a continuous function.

Since the plasma protein concentration can be expressed in terms of itsmass and plasma volume as

${c_{p}(t)} = \frac{m_{p}(t)}{V_{p}(t)}$

and the change of protein mass in the plasma at time t is determined bythe net protein flux as

$\begin{matrix}{{\frac{d\; {m_{p}(t)}}{dt} = {J_{s}(t)}},} & (7)\end{matrix}$

the change in plasma protein concentration is obtained as

$\begin{matrix}{\frac{d\; {c_{p}(t)}}{dt} = {\frac{{J_{s}(t)} - {{c_{p}(t)}\frac{{dV}_{p}(t)}{dt}}}{V_{p}(t)}.}} & (8)\end{matrix}$

The change in interstitial volume is governed by the volume lost toplasma and the lymphatic system and thus,

$\begin{matrix}{\frac{{dV}_{i}(t)}{dt} = {{J_{v}(t)} - {\kappa.}}} & (9)\end{matrix}$

By a similar argument, the mass of proteins that goes to the plasmacompartment is the loss term in the interstitium compartment and thusthe change of the interstitial protein mass is

$\begin{matrix}{{\frac{d\; {m_{i}(t)}}{dt} = {- {J_{s}(t)}}},} & (10)\end{matrix}$

and with interstitial protein concentration as

${c_{i}(t)} = \frac{m_{i}(t)}{V_{i}(t)}$

the change in interstitial protein concentration is given by

$\begin{matrix}{\frac{d\; {c_{i}(t)}}{dt} = {\frac{{- {J_{s}(t)}} + {{c_{i}(t)}\left( {{J_{v}(t)} + \kappa} \right)}}{V_{i}(t)}.}} & (11)\end{matrix}$

Model Equations:

The dynamics of the two-compartment model is described by the followingsystem of ordinary differential equations

$\begin{matrix}\left\{ {{{\begin{matrix}{{\frac{{dV}_{p}}{dt} = {J_{v} + \kappa - J_{UF}}},} \\{{\frac{d\; c_{p}}{dt} = \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}}},} \\{{\frac{{dV}_{i}}{dt} = {{- J_{v}} - \kappa}},} \\{{\frac{d\; c_{i}}{dt} = \frac{{- J_{s}} + {c_{i}\left( {{J_{v}(t)} + \kappa} \right)}}{V_{i}}},}\end{matrix}{where}J_{v}} = {L_{p}\left( {{\sigma \left( {\left( {{a_{p_{1}}c_{p}} + {a_{p_{2}}c_{p}^{2}}} \right) - \left( {{a_{i_{1}}c_{i}} + {a_{i_{2}}c_{i}^{2}}} \right)} \right)} - \left( {P_{c} - P_{i}} \right)} \right)}},{{J_{s}(t)} = \left\{ {{\begin{matrix}{{{J_{v}\left( {1 - \sigma} \right)}\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1}} \right)} + {\alpha \; \kappa}} & {{{{if}\mspace{14mu} J_{v}} > 0},} \\{\alpha\kappa} & {{{{if}\mspace{14mu} J_{v}} = 0},} \\{{{J_{v}\left( {1 - \sigma} \right)}\left( {c_{p} - \frac{c_{p} - c_{i}}{e^{x} - 1}} \right)} + {\alpha \; \kappa}} & {{{{if}\mspace{14mu} J_{v}} < 0},}\end{matrix}{and}x} = {\frac{J_{v}\left( {1 - \sigma} \right)}{PS}.}} \right.}} \right. & (12)\end{matrix}$

Model Output:

In order to estimate certain parameters, the model is compared tomeasurement data. The Crit-Line Monitor device is an exemplary devicethat provides readings of the hematocrit concentration and oxygensaturation during hemodialysis. It is a non-invasive method based on anoptical sensor technique. The sensor is attached to a blood chamber andis placed in-line between the arterial blood tubing set and thedialyzer. The measurements are based on both the absorption propertiesof the hemoglobin molecule and the scattering properties of red bloodcells. Hematocrit levels can be expressed in terms of the model statevariables, namely, V_(p), c_(p), V_(i), and c_(i) for parameteridentification.

Let BV(t) and V_(p)(t) denote the blood volume and the plasma volume,respectively, at time t. Note that

V _(p)(t)−V _(p)(0)=BV(t)−BV(0).  (13)

Expressing the blood volume in terms of plasma volume and the hematocritin Eq. (13) and rearranging the terms yields

${{V_{p}(t)} - {V_{p}(0)}} = {\frac{V_{p}(t)}{1 - {{Hct}(t)}} - \frac{V_{p}(0)}{1 - {{Hct}(0)}}}$${{Hct}(t)} = \frac{{\left( {1 - {{Hct}(0)}} \right){V_{p}(t)}} + {{{Hct}(0)}{V_{p}(0)}} - {\left( {1 - {{Hct}(0)}} \right){V_{p}(t)}}}{{\left( {1 - {{Hct}(0)}} \right){V_{p}(t)}} + {{{Hct}(0)}{V_{p}(0)}}}$

Therefore, Hct at time t can be expressed in terms of initialhematocrit, initial plasma volume and V_(p) at time t as follows

$\begin{matrix}{{{{Hct}(t)} = {\frac{{{Hct}(0)}{V_{p}(0)}}{{\left( {1 - {{Hct}(0)}} \right){V_{p}(t)}} + {{{Hct}(0)}{V_{p}(0)}}} = \frac{1}{{H_{0}{V_{p}(t)}} + 1}}}{where}} & (14) \\{H_{0} = {\frac{1 - {{Hct}(0)}}{{{Hct}(0)}{V_{p}(0)}}.}} & (15)\end{matrix}$

Simulations:

First, some theoretical results are presented assigning values to theparameters found in the literature. Table 1 provides the list ofparameters, its meaning, corresponding values and units used in themodel.

For model simulation, the following initial conditions are considered:

TABLE 1 Parameter values Parameter Meaning Value Range Unit L_(p)filtration coefficient  1.65 1.65 ± 1.92 mL/mmHg/min σ systemiccapillary reflection  0.9 0.75 − 0.95 coefficient P_(c) hydrostaticcapillary pressure 21.1 21.1 ± 4.9 mmHg P_(i) hydrostatic interstitialpressure  2 −1.5 − 4.6 mmHg α_(p) ₁ coefficient of c_(p) in Eq. (3) 0.1752 mmHg(mL/mg) α_(p) ₂ coefficient of c_(p) ² in Eq. (3)  0.0028mmHg(mL/mg)² α_(i) ₁ coefficient of c_(i) in Eq. (3)  0.2336 mmHg(mL/mg)α_(i) ₂ coefficient of c_(i) ² in Eq. (3)  0.0034 mmHg(mL/mg)² PSpermeability surface area product  0.45 m/min κ constant lymph flow rate 1.5 1.39 − 2.78 mL/min J_(UF) ultra filtration rate 15 (900 mL/hour)mL/min

-   -   Initial plasma colloid osmotic pressure π_(p) is known from        which initial plasma protein concentration c_(p) is obtained        using Eq. (3). See Appendix B.    -   Initial interstitial colloid osmotic pressure π_(i),        interstitial protein concentration c_(i) and constant protein        concentration from the lymph flow α are computed assuming        equilibrium prior to ultrafiltration, that is, lymphatic flow        balances capillary filtration. See Appendix B.    -   Initial interstitial volume V_(i) is 4.3 times initial plasma        volume V_(p), that is, V_(i)=4.3 V_(p). This is based on data        from dialysis patients.

The predialysis plasma colloid osmotic pressure of π_(p)*=28 mmHg hasbeen reported. This value is used to compute the initial c_(p) andc_(i). The computed value for the protein concentration assumingequilibrium is α=24.612. Initial plasma volume is set at 4000 mL and theinitial interstitial volume is calculated based on the volume relationmentioned above. The initial values for the state variables are listedin Table 2.

TABLE 2 Computed equilibrium/initial values State Meaning Value UnitV_(p) ⁰ initial plasma volume  4.000 mL c_(p) ⁰ initial plasma proteinconcentration 73.4040 mg/mL V_(i) ⁰ initial interstitial volume  1.7200mL c_(i) ⁰ initial interstitial protein concentration 24.4153 MG/mL

The period of one hour is divided into three phases, namely: rest phase,ultrafiltration phase, and refill phase. During rest and refill phases,J_(F) is set to 0 and during ultrafiltration (UF) phase, J_(UF) is setabove the regular UF rate. FIG. 8 illustrates the dynamical behavior ofthe model state variables for an hour where J_(UF)=30 mL/min for 20minutes on t∈[20, 40] and J_(UF)=0 for t∈[0, 20) and t∈(40,60]. Theupper left panel shows that V_(p) decreases during fluid removal. Whenthe ultrafiltration is turned oft an increase in V_(p) is observedsignifying the movement of fluid from the interstitium to the plasmawhich indicates vascular refilling. On the upper right panel, V_(i)decreases during the UF and refilling phases even when there is noultrafiltration. Thus, fluid continues to move from interstitium toplasma and hence a fluid loss in this compartment. The bottom paneldepicts the dynamics of plasma and interstitial protein concentrationsduring the given intervention. Notice that c_(p) increases duringultrafiltration and decreases slightly during refill phase while c_(i)does not change significantly. Overall, the model dynamics reflect thequalitative physiological behavior as one would expect during ashort-pulse ultrafiltration.

Hematocrit is initially set at Hct(0)=22 and then Eq. (14) is used toobtain a plot for the model output. FIG. 9 illustrates hematocrit levelsas model output during a rest phase for t∈[0, 20) minutes, UF atJ_(UF)=30 mL/min for t∈[20, 40] minutes, and refill phase for t∈(40, 60]minutes. As expected, hematocrit level increases during ultrafiltrationsince it is assumed that the red blood cell mass does not change whilefluid is removed.

Sensitivity Analysis and Subset Selection:

Sensitivity analysis and subset selection provide insights on theinfluence of certain parameters on the model output and on theidentifiability of parameters with regard to specific measurements.Further, the information of these analyses can be used for experimentaldesign. It helps in making informed decisions on the type ofmeasurements, the frequency and the precision of the specificmeasurements needed to identify parameters. In the context of thisapplication, it is important to ensure that with the gathered data it isindeed possible to identify the parameters we are interested in.

Traditional and Generalized Sensitivity Functions:

In simulation studies, the traditional sensitivity functions (TSFs) arefrequently used to assess the degree of sensitivity of a model outputwith respect to various parameters on which it depends. It shows whatparameters influence the model output the most or the least. The moreinfluential changes in a parameter on the model output, the moreimportant it is to assign accurate values to that parameter. On theother hand, for parameters which are less sensitive it suffices to havea rough estimate for the value. To obtain information content on theparameters, the model output needs to be either a state variable of thesystem or expressible as one (or more) of the state variables. Here, arelationship is established between the model output (hematocrit) and astate variable (plasma volume V_(p)) (see above).

The generalized sensitivity functions (GSFs) provide information on thesensitivity of parameter estimates with respect to model parameters. Itdescribes the sensitivity of parameter estimates with respect to theobservations or specific measurements. Note, it is assumed that themeasurement error is normally distributed with a given standarddeviation. Some details on sensitivities are provided in Appendix C.Based on the GSFs, one can assess if the parameters of interest arewell-identifiable assuming a priori measurement error, measurementfrequency and nominal parameter values.

Sensitivity Equations for the Vascular Refill Model:

Let y(t)=Hct(t), i.e. the hematocrit level at time t is defined as themodel output (see Eq. (14)). The sensitivity of the model output withrespect to the parameter L can be determined as follows

$\begin{matrix}{{{}_{}^{}{}_{}^{\;}} = {\frac{L_{p}}{y(t)}\frac{\partial{y(t)}}{\partial L_{p}}}} \\{= {\frac{L_{p}}{\frac{1}{{H_{0}{V_{p}(t)}} + 1}}\left( \frac{- H_{0}}{\left( {{H_{0}{V_{p}(t)}} + 1} \right)^{2}} \right)\frac{\partial{V_{p}(t)}}{\partial L_{p}}}}\end{matrix}$

which can be simplified as

$\begin{matrix}{{{{}_{}^{}{}_{}^{\;}} = {{- \frac{L_{p}H_{0}}{{H_{0}{V_{p}(t)}} + 1}}\frac{\partial{V_{p}(t)}}{\partial L_{p}}}}{where}{H_{0} = {\frac{1 - {{Hct}(0)}}{{{Hct}(0)}{V_{p}(0)}}.}}} & (16)\end{matrix}$

Sensitivities with respect to other parameters can be obtained similarlyand they are given as

${s_{\sigma} = {{\frac{\sigma}{y(t)}\frac{\partial{y(t)}}{\partial\sigma}} = {{- \frac{{\sigma H}_{0}}{{H_{0}{V_{p}(t)}} + 1}}\frac{\partial V_{p}}{\partial\sigma}}}},{s_{PS} = {{\frac{PS}{y(t)}\frac{\partial{y(t)}}{\partial{PS}}} = {\frac{{PSH}_{0}}{{H_{0}{V_{p}(t)}} + 1}\frac{\partial V_{p}}{\partial{PS}}}}},{s_{Pc} = {{\frac{P_{c}}{y(t)}\frac{\partial{y(t)}}{\partial P_{c}}} = {\frac{P_{c}H_{0}}{{H_{0}{V_{p}(t)}} + 1}\frac{\partial V_{p}}{\partial P_{c}}}}},{s_{Pi} = {{\frac{P_{i}}{y(t)}\frac{\partial{y(t)}}{\partial P_{i}}} = {\frac{P_{i}H_{0}}{{H_{0}{V_{p}(t)}} + 1}\frac{\partial V_{p}}{\partial P_{i}}}}},{s_{a_{p\; 1}} = {{\frac{P_{p_{1}}}{y(t)}\frac{\partial{y(t)}}{\partial a_{p\; 1}}} = {\frac{P_{p_{1}}H_{0}}{{H_{0}{V_{p}(t)}} + 1}\frac{\partial V_{p}}{\partial P_{p_{1}}}}}},{s_{a_{p\; 2}} = {{\frac{P_{p_{2}}}{y(t)}\frac{\partial{y(t)}}{\partial a_{p\; 2}}} = {\frac{P_{p_{2}}H_{0}}{{H_{0}{V_{p}(t)}} + 1}\frac{\partial V_{p}}{\partial P_{p_{2}}}}}},{s_{a_{i\; 1}} = {{\frac{P_{i_{1}}}{y(t)}\frac{\partial{y(t)}}{\partial a_{i\; 1}}} = {\frac{P_{i_{1}}H_{0}}{{H_{0}{V_{p}(t)}} + 1}\frac{\partial V_{p}}{\partial P_{i_{1}}}}}},{s_{a_{i\; 2}} = {{\frac{P_{i_{2}}}{y(t)}\frac{\partial{y(t)}}{\partial a_{i\; 2}}} = {\frac{P_{i_{2}}H_{0}}{{H_{0}{V_{p}(t)}} + 1}\frac{\partial V_{p}}{\partial P_{i_{2}}}}}},{s_{\kappa} = {{\frac{\kappa}{y(t)}\frac{\partial{y(t)}}{\partial\kappa}} = {{- \frac{\kappa \; H_{0}}{{H_{0}{V_{p}(t)}} + 1}}\frac{\partial V_{p}}{\partial\kappa}}}},{s_{\alpha} = {{\frac{\kappa}{y(t)}\frac{\partial{y(t)}}{\partial\alpha}} = {{- \frac{\alpha \; H_{0}}{{H_{0}{V_{p}(t)}} + 1}}\frac{\partial V_{p}}{\partial\alpha}}}},{s_{J_{UF}} = {{\frac{J_{UF}}{y(t)}\frac{\partial{y(t)}}{\partial\kappa}} = {{- \frac{\kappa \; H_{0}}{{H_{0}{V_{p}(t)}} + 1}}{\frac{\partial V_{p}}{\partial\kappa}.}}}}$

The derivatives of the states with respect to the parameters∂x(t)/∂p_(k), ∂V_(p)(t)/∂L_(p), etc. can be found in Appendix C.4.

FIGS. 10 and 11 are plots illustrating traditional sensitivities ofmodel output with respect to certain parameters. The magnitude of TSFsdetermines how sensitive the model output is to a specific parameter ina given time interval. That is, TSFs with greater magnitude have highersensitivities in a certain period. Plots of TSFs corresponding to therest, UF and refill phases discussed above are shown in FIG. 10. It canbe seen that L_(p) and σ have high sensitivities. Thus, it can beexpected that a unit change in these parameters will have a significantinfluence in the model output dynamics compared to a unit change inother parameters. FIG. 11 depicts the TSFs of parameters with smallermagnitude only. It also illustrates that parameter L_(p) becomes moresensitive on certain times.

Subset Selection:

Before an actual parameter identification is carried out, one can choosea priori which parameters can be estimated given a data set. A subsetselection algorithm described in Appendix C.3 and in Cintrón-Arias A,Banks H T, Capaldi A, Lloyd A L “A sensitivity matrix based methodologyfor inverse problem formulation” J Inv Ill-posed Problems 17:545-564(2009), the entirety of which is incorporated by reference herein,chooses the parameter vectors that can be estimated from a given set ofmeasurements using an ordinary least squares inverse problemformulation. The algorithm requires prior knowledge of a nominal set ofvalues for all parameters along with the observation times for data.Among the given set of parameters, the algorithm searches all possiblechoices of different parameters and selects those which areidentifiable. It minimizes a given uncertainty quantification, forinstance, by means of asymptotic standard errors. Overall, subsetselection gives information on local identifiability of a parameter setfor a given data. Further, it gives a quantification whether a parametercan be identified or not.

The subset selection algorithm is used to select best combination ofparameters from the given set of model parameters based on a definedcriteria. As mentioned above, prior knowledge of measurement varianceσ₀, measurement frequency and nominal parameter values θ₀ are required.These values are needed to compute the sensitivity matrix, the FisherInformation Matrix (FIM) and the corresponding covariance matrix. In thecurrent study, a selection score α(θ₀) is set to be the maximum norm ofthe coefficients of variation for θ (see Appendix C.3 for more details).The subset of parameters are chosen with the minimal α(θ₀). Thecondition number cond(F(θ₀)) determines the ratio between the largestand the smallest eigenvalue of the FIM. If cond(F(θ₀)) is too large, FIMis ill-posed and the chosen subset of parameters might be difficult toidentify or even unidentifiable.

Table 3 presents the chosen parameters out of the 12 model parameterswith the selection score and condition number of the corresponding FIM.It is assumed that measurement can be obtained at the frequency of 10Hz, the standard error is 0.1 (variance is 0.01) and the nominalparameter values are given in Table 1. Note that L_(p), P_(c), and σ arethe three best parameter combinations chosen. This method suggests thatthese parameters can be identified given the measurements withproperties mentioned earlier.

TABLE 3 Subset selection choosing from 12 parameters (L_(p), σ, PS,P_(c), P_(i), κ, α,) No. Parameter vector θ α (θ₀) cond (F (θ₀) 2(L_(p), P_(c)) 3.05 × 10⁻⁵ 5.3712 3 (L_(p), σ, P_(c)) 0.00040641 3.0162× 10⁷ 4 (σ, α_(p) ₂ , α_(i) ₂ , α) 0.003508 2.0462 × 10¹⁴ (L_(p), , σ,Pc, α) 0.0036051  3.812 × 10⁹ 5 (L_(p), σ, α_(p) ₁ , α_(i) ₂ , α)0.024172 2.6421 × 10¹⁴

Since hematocrit is a measurement that can be obtained using theCrit-Line Monitor device (among other ways), some parameters need not beestimated using this observation. Specifically, a_(p1), a_(p2), a_(i1),a_(i2) are not in top priority to be identified because proteinconcentration measurements might be necessary for this purpose. Also,the ultrafiltration rate J_(UF) is an outside perturbation introduced inthe system which can be set a priori. Table 4 shows the parameterselection of the algorithm choosing from 7 parameters. It is importantto note that L_(p), P_(c), σ are selected which are of significantrelevance in this example.

TABLE 4 Subset selection choosing from 7 parameters (L_(p), σ, PS,P_(c), P_(i), κ, α,) No. Parameter vector θ α (θ₀) cond (F (θ₀) 2(L_(p), P_(c)) 3.05 × 10⁻⁵ 5.3712 3 (L_(p), σ, P_(c)) 0.00040641 3.0162× 10⁷ 4 (L_(p), σ, P_(c), κ) 0.0053168 1.0971 × 10⁸ 5 (L_(p), σ, PS,P_(c), κ) 1.783 5.1709 × 10¹²

Model Identification:

Model identifiability is assessed to determine parameters from measureddata. The term refers to the issue of ascertaining unambiguous andaccurate parameter estimation. Parameter estimation determines aparameter set such that the model output is as close as possible to thecorresponding set of observations. This accounts for minimizing ameasure of error for the difference between model output andmeasurements. It should be noted that the quality of the parameterestimates depends on the error criterion, model structure and fidelityof the available data.

To test the adaptability of the current model, a patient's hematocritdata is used. Though the measurement obtained with varyingultrafiltration profiles was originally collected for a differentpurpose, it can be shown that the present model adapts to the given setof observations despite this limitation. In particular, some keyparameters can be identified.

Parameter Estimation for Patient 1 Data:

FIG. 12 is a graph illustrating an exemplary model output where themodel is adapted to the hematocrit measurements of a specific patient,patient “1” (black curve), by identifying L_(p) and P_(c). Theparameters were estimated using measurements within the two verticaldashed lines and hematocrit values were predicted for the following 20minutes (white curve). The white curve is the model output Eq. (14)obtained by solving the system of ordinary differential equations givenin Eq. (12). The model has a good prediction for the next 20 minutesafter the estimation. Hence, the model can predict the dynamics ofvascular refilling for a short period of time.

Table 5 indicates that for this particular patient data set, L_(p) andP_(c) are identifiable. As shown, varied initial parameter valuesconverge to the same estimated values (to some degree of accuracy). Itindicates local identifiability of these model parameters.

TABLE 5 Identification of (L_(p), P_(c)) Initial Value Estimated Value(1.65, 21.1) (2.8945, 20.4169) (1.7325, 22.1550) (2.8928, 20.4178)(1.8150, 23.2100) (2.8952, 20.4186) (1.5675, 20.0450) (2.8929, 20.4175)(1.4850, 18.9900) (2.8961, 20.4183) (1.4850, 18.9900) (2.8962, 20.4184)(2.8, 20) (2.8962, 20.4184) (3.0800, 22) (2.8930, 20.4176) (2.5200, 18)(2.8927, 20.4169)

Parameter Estimation for Patient 2 Data:

Here, data of a different, second patient “2” is used to illustrate thevalidity of the model. In this case, three parameters, namely, L_(p),P_(c), and κ are estimated. FIG. 13 illustrates the second patient dataand the corresponding parameter identification and model prediction. Asin the previous illustration, the parameters of interest are identifiedfrom the data for t∈[10, 50]. It can be seen that the model with theestimated parameters provide a good prediction for the next 20 minutes.

Table 6 shows that different initial values of the parameters L_(p),P_(c), and κ converge to the same estimated values. Thus, it indicateslocal identifiability of these parameters.

TABLE 6 Identification of (L_(p), P_(c), κ). Initial Value EstimatedValue (30, 20, 10) (30.3206, 21.2392, 10.3555) (30.3, 21.2, 10.3)(30.3435, 21.2389, 10.3529) (31.8150, 22.26, 10.815) (30.3178, 21.2395,10.3590) (33.3, 23,32, 11.33) (30.3365, 21.2392, 10.3567) (29.5425,20.67, 10.0495) (30.3095, 21.2393, 10.3553) (27.27, 19.08, 9.97)(30.3326, 21.2387, 10.3505)

All references, including publications, patent applications, andpatents, cited herein are hereby incorporated by reference to the sameextent as if each reference were individually and specifically indicatedto be incorporated by reference and were set forth in its entiretyherein.

The use of the terms “a” and “an” and “the” and “at least one” andsimilar referents in the context of describing the invention (especiallyin the context of the following claims) are to be construed to coverboth the singular and the plural, unless otherwise indicated herein orclearly contradicted by context. The use of the term “at least one”followed by a list of one or more items (for example, “at least one of Aand B”) is to be construed to mean one item selected from the listeditems (A or B) or any combination of two or more of the listed items (Aand B), unless otherwise indicated herein or clearly contradicted bycontext. The terms “comprising,” “having,” “including,” and “containing”are to be construed as open-ended terms (i.e., meaning “including, butnot limited to,”) unless otherwise noted. Recitation of ranges of valuesherein are merely intended to serve as a shorthand method of referringindividually to each separate value falling within the range, unlessotherwise indicated herein, and each separate value is incorporated intothe specification as if it were individually recited herein. All methodsdescribed herein can be performed in any suitable order unless otherwiseindicated herein or otherwise clearly contradicted by context. The useof any and all examples, or exemplary language (e.g., “such as”)provided herein, is intended merely to better illuminate the inventionand does not pose a limitation on the scope of the invention unlessotherwise claimed. No language in the specification should be construedas indicating any non-claimed element as essential to the practice ofthe invention.

Preferred embodiments of this invention are described herein, includingthe best mode known to the inventors for carrying out the invention.Variations of those preferred embodiments may become apparent to thoseof ordinary skill in the art upon reading the foregoing description. Theinventors expect skilled artisans to employ such variations asappropriate, and the inventors intend for the invention to be practicedotherwise than as specifically described herein. Accordingly, thisinvention includes all modifications and equivalents of the subjectmatter recited in the claims appended hereto as permitted by applicablelaw. Moreover, any combination of the above-described elements in allpossible variations thereof is encompassed by the invention unlessotherwise indicated herein or otherwise clearly contradicted by context.

APPENDIX A Quadratic Approximation to Colloid Osmotic Pressures

The colloid osmotic pressure π_(p), π_(i) in plasma and in theinterstitium can be expressed in terms of its respective proteinconcentrations c_(p), c_(i) as follows

π_(p)=α_(p) ₁ c _(p)+α_(p) ₂ c _(p) ²+α_(p) ₃ c _(p) ³ , c _(p)≥0,

π_(i)=α_(i) ₁ c _(i)+α_(i) ₂ c _(i) ²+α_(i) ₃ c _(i) ³ , c _(i)≥0,

The coefficients are given by

-   -   α_(p) ₁ =0.21, α_(p) ₂ =0.0016, α_(p) ₃ =0.000009,    -   α_(i) ₁ =0.28, α_(i) ₂ =0.0018, α_(i) ₃ =0.000012,        the units being mmHg(mL/mg), mmHg(mL/mg)² and mmHg(mL/mg)³. The        plot for π_(p) and π_(i) shown in FIG. 14 for 0≤c_(p), c_(i)≤100        mg/mL indicates that quadratic polynomials instead of cubic        polynomials would capture the relevant dynamics using fewer        parameters.

The quadratic approximations π_(p.approx)(c_(p))=α₁c_(p)+α₂c_(p) ² andπ_(i,approx)(c_(i))=β₁c_(i)+β₂c_(i) ² of π_(p)(c_(p)) and π_(i)(c_(i))are computed by minimizing

${{{{\pi_{p}( \cdot )} - {\pi_{p,{approx}}( \cdot )}}}_{\mathcal{L}^{1}} = {\sum\limits_{j = 0}^{1000}{{{\pi_{p}\left( {0.1j} \right)} - {\pi_{p,{approx}}\left( {0.1j} \right)}}}}},{{{{\pi_{i}( \cdot )} - {\pi_{p,{approx}}( \cdot )}}}_{\mathcal{L}^{1}} = {\sum\limits_{j = 0}^{1000}{{{\pi_{i}\left( {0.1j} \right)} - {\pi_{i,{approx}}\left( {0.1j} \right)}}}}},$

The obtained coefficients α_(k), β_(k), k=1, 2, are given by

-   -   α₁=0.1752, α₂=0.0028, β₁=0.2336, β₂=0.0034.

In FIG. 15 we show the function π_(p) and π_(i) together with theapproximating polynomials π_(p,approx) and π_(i,approx), whereas in FIG.16 we present the differences π_(p)−π_(p,approx) and π_(i)−π_(i,approx).The maximal errors occur at c_(p)=c_(i)=100 and are given by 0.7774 forπ_(p) and 1.0346 for π_(i).

B Computation of Equilibria

Let the colloid osmotic pressures π_(p), π_(i) in the plasma and theinterstitium be given as

π_(p)=α_(p) ₁ c _(p)+α_(p) ₂ c _(p) ² , c _(p)≥0,

π_(i)=α_(i) ₁ c _(i)+α_(i) ₂ c _(i) ² , c _(i)≥0,

where c_(p) respectively c_(i) is the protein concentration in plasmarespectively in the interstitium. Assume that the equilibrium π_(p)*value is known. The equilibrium c_(p) can be computed by solving thequadratic equation

a _(p) ₂ (c _(p)*)²+α_(p) ₁ c _(p)*−π_(p)*=0.

Using quadratic formula, the roots of the above equation are

$c_{p}^{*} = {\frac{{- a_{p_{1}}} \pm \sqrt{a_{p_{1}}^{2} + {4a_{p_{2}}\pi_{p}^{*}}}}{2\; a_{p_{2}}}.}$

There exists a real root provided that the discriminant α_(p) ₁ ²+4α_(p)₂ π_(p)*>0. Hence, to ensure that c_(p)* is positive, the followingequation has to be satisfied

−α_(p) ₁ +√{square root over (α_(p) ₁ ²+4α_(p) ₂ π_(p)*)}>0,

which is trivially satisfied since π_(p)* is always positive.

Assuming all the parameters are known including the constant lymph flowto the plasma κ, the equilibrium interstitial colloid osmotic pressurec_(i)* can be obtained by solving the equilibria of our model. That is,we have

$\begin{matrix}{{J_{v} = {- \kappa}},} \\{{{L_{p}\left( {{\sigma \left( {\pi_{p}^{*} - \pi_{i}^{*}} \right)} - \left( {P_{c} - P_{i}} \right)} \right)} = {- \kappa}},} \\{\pi_{i}^{*} = {\pi_{p}^{*} - {\frac{1}{\sigma}{\left( {{- \frac{\kappa}{L_{f}p}} + \left( {P_{c} - P_{i}} \right)} \right).}}}}\end{matrix}$

Expressing the π_(i)* in terms of c_(i) yields

${{a_{i_{1}}c_{i}^{*}} + {a_{i_{2}}\left( c_{i}^{*} \right)}^{2}} = {\pi_{p}^{*} - {\frac{1}{\sigma}{\left( {{- \frac{\kappa}{L_{p}}} + \left( {P_{c} - P_{i}} \right)} \right).}}}$

As above, in order to obtain a real positive c_(i)*, the followingequation needs to be satisfied

${{- a_{i_{1}}} + \sqrt{a_{i_{1}}^{2} + {4{a_{i_{2}}\left( {\pi_{p}^{*} - {\frac{1}{\sigma}\left( {{- \frac{\kappa}{L_{p}}} + \left( {P_{c} - P_{i}} \right)} \right)}} \right)}}}} > 0.$

The equilibrium value for c_(i) is then given by

$c_{i}^{*} = {\frac{{- a_{i_{1}}} + \sqrt{a_{i_{1}}^{2} + {4{a_{i_{2}}\left( {\pi_{p}^{*} - {\frac{1}{\sigma}\left( {{- \frac{\kappa}{L_{p}}} + \left( {P_{c} - P_{i}} \right)} \right)}} \right)}}}}{2a_{i_{2}}}.}$

C Sensitivities C.1 Traditional Sensitivities

Let the variable y=y(θ) for θ∈D, where D is some open interval andassume that y is differentiable on D. Let θ₀∈D be given and assume thatθ₀≠0 and y₀=y(θ₀)≠0. Here, θ₀ denotes the initial/nominal parameter andy₀ refers to the initial model output. The sensitivity s_(y)(θ₀) of ywith respect to θ at θ₀ is defined as:

${s_{y,\theta}\left( \theta_{0} \right)} = {{\lim\limits_{{\Delta\theta}->0}\frac{\Delta \; {y/y_{0}}}{\Delta/\theta_{0}}} = {\frac{\theta_{0}}{y_{0}} = {\frac{\theta_{0}}{y_{0}}{{y^{\prime}\left( \theta_{0} \right)}.}}}}$

The sensitivities s_(x,θ)(θ₀) are defined such that they are invariantagainst changes of units in both θ and y.

In general, we have a dynamical system of the form

$({AWP})\left\{ \begin{matrix}{{{\overset{.}{x}(t)} = {\mathcal{F}\left( {t,{x(t)},\theta} \right)}},} \\{{{x(0)} = {x_{0}(\theta)}},}\end{matrix} \right.$

where x(t) is the vector of state variables of the system, θ is thevector of system parameters and t∈[0, T]. We define

y(t)=f(t,θ), 0≤t≤T,

to be the (single) output of the system.

In order to compute the traditional sensitivity functions (TSF)(sensitivity of a model output with respect to various parameters onwhich it depends) as well as for the generalized sensitivities (GSF)(sensitivity of parameter estimates with respect to measurements) the socalled sensitivity equations are needed. The sensitivity equations are alinear ODE-system of the following form

${{\overset{.}{S}\left( {t,\theta} \right)} = {{{\mathcal{F}_{x}\left( {t,{x\left( {t,\theta} \right)},\theta} \right)}{S\left( {t,\theta} \right)}} + {F_{\theta}\left( {t,{x\left( {t,\theta} \right)},\theta} \right)}}},{{S\left( {0,\theta} \right)} = \frac{\partial{x_{0}(\theta)}}{\partial\theta}},$

where

_(x)(·) and

_(θ)(·) denote the partial derivative of

with respect to the state variable x and parameter θ, respectively.Equation (17) in conjunction with (AWP) provides a fast and sufficientway to compute the sensitivity matrix S(t,θ) numerically.

C.2 Generalized Sensitivities

The generalized sensitivity function g_(S)(t_(l)) with respect to theparameter θ_(k) at the time instant t_(l) for θ in a neighborhood of θ₀(the initial/nominal parameter vector) is given by

${{g_{S}\left( t_{l} \right)} = {\sum\limits_{i = 1}^{l}{\frac{1}{\sigma^{2}\left( t_{i} \right)}\left( {F^{- 1}{\nabla_{\theta}{f\left( {t_{i},\theta} \right)}}} \right){\nabla_{\theta}{f\left( {t_{i},\theta} \right)}}}}},$

and the Fisher Information Matrix F is given by

${F = {\sum\limits_{j = 1}^{N}{\frac{1}{\sigma^{2}\left( t_{j} \right)}{\nabla_{\theta}{f\left( {t_{j},\theta} \right)}}{\nabla_{\theta}{f\left( {t_{j},\theta} \right)}^{T}}}}},$

where t₁, . . . , t_(N) denotes the measurement points.

C.3 Subset Selection Algorithm

Given p<p₀, the algorithm taken from the literature considers allpossible indices i₁, . . . , i_(p) with 1≤i_(i)< . . . <i_(p)≤p₀ inlexicographical ordering starting with the first choice (i₁ ⁽¹⁾, . . . ,i_(p) ⁽¹⁾)=(1, . . . , p) and completes the following steps:

Initializing step: Set ind^(sel)=(1, . . . , p) and α^(sel)=∞.

Step k: For the choice (i₁ ^((k)), . . . , i_(p) ^((k))) compute r=rankF((q₀)_(i) ₁ _((k)) , . . . , (q₀)_(i) _(p) _((k)) ).

-   -   If r<p, go to Step k+1.    -   If r=p, compute α_(k)=((q₀)_(i) ₁ _((k)) , . . . . , (q₀)_(i)        _(p) _((k)) ).        -   If α_(k)≥α^(sel), go to Step k+1.        -   If α_(k)<α, set ind^(sel)=(i₁ ^((k)), . . . , i_(p) ^((k)),            α^(sel)=α_(k) and go to Step k+1.

F is the Fisher information matrix mentioned in the preceding section.

C.4 Sensitivity with Respect to a Parameter

It can be easily verified that the partial derivative of J_(v) withrespect to c_(p) and c_(i) are

$\begin{matrix}{{\frac{\partial J_{v}}{\partial c_{p}} = {L_{p}{\sigma \left( {a_{p_{1}} + {2a_{p_{2}}c_{p}}} \right)}}},{\frac{\partial J_{v}}{\partial c_{i}} = {{- L_{p}}{\sigma \left( {a_{i_{1}} + {2a_{i_{2}}c_{i}}} \right)}}},} & (18)\end{matrix}$

respectively. Also, the following can be obtained immediately

$\begin{matrix}{{{\frac{\partial}{\partial c_{p}}\left( {e^{x} - 1} \right)} = {e^{x}\frac{1 - \sigma}{PS}\frac{\partial J_{v}}{\partial c_{p}}}},{{\frac{\partial}{\partial c_{i}}\left( {e^{x} - 1} \right)} = {e^{x}\frac{1 - \sigma}{PS}\; {\frac{\partial J_{v}}{\partial c_{i}}.}}}} & (19)\end{matrix}$

The partial derivative of J_(s) with respect to c_(p) when J_(v)>0 canbe derived as follows

$\begin{matrix}{\frac{\partial J_{s}}{\partial c_{p}} = {\left( {1 - \sigma} \right)\left( {{\frac{\partial J_{v}}{\partial c_{p}}\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1}} \right)} - {J_{v}\left( \frac{\left( {e^{x} - 1} \right) - {\left( {c_{p} - c_{i}} \right)e^{x}\frac{1 - \sigma}{PS}\frac{\partial J_{v}}{\partial c_{p}}}}{\left( {e^{x} - 1} \right)^{2}} \right)}} \right)}} \\{= {\left( {1 - \sigma} \right)\left( {{\frac{\partial J_{v}}{\partial c_{p}}\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1}} \right)} - \left( \frac{{J_{v}\left( {e^{x} - 1} \right)} - {\left( {c_{p} - c_{i}} \right)e^{x}\frac{J_{v}\left( {1 - \sigma} \right)}{PS}\frac{\partial J_{v}}{\partial c_{p}}}}{\left( {e^{x} - 1} \right)^{2}} \right)} \right)}} \\{{= {\left( {1 - \sigma} \right)\left( {{\frac{\partial J_{v}}{\partial c_{p}}\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1}} \right)} - \left( \frac{{J_{v}\left( {e^{x} - 1} \right)} - {\left( {c_{p} - c_{i}} \right){xe}^{x}\frac{\partial J_{v}}{\partial c_{p}}}}{\left( {e^{x} - 1} \right)^{2}} \right)} \right)}},} \\{\frac{\partial J_{s}}{\partial c_{p}} = {\left( {1 - \sigma} \right){\left( {{\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1} + \frac{\left( {c_{p} - c_{i}} \right){xe}^{x}}{\left( {e^{x} - 1} \right)^{2}}} \right)\frac{\partial J_{v}}{\partial c_{p}}} - \frac{J_{v}}{e^{x} - 1}} \right).}}}\end{matrix}$

When J_(v)<0, we have

$\begin{matrix}{\frac{\partial J_{s}}{\partial c_{p}} = {\left( {1 - \sigma} \right)\left( {{\frac{\partial J_{v}}{\partial c_{p}}\left( {c_{p} - \frac{c_{p} - c_{i}}{e^{x} - 1}} \right)} + {J_{v}\left( {1 - \frac{\left( {e^{x} - 1} \right) - {\left( {c_{p} - c_{i}} \right)e^{x}\frac{1 - \sigma}{PS}\frac{\partial J_{v}}{\partial c_{p}}}}{\left( {e^{x} - 1} \right)^{2}}} \right)}} \right)}} \\{= {\left( {1 - \sigma} \right)\left( {{\frac{\partial J_{v}}{\partial c_{p}}\left( {c_{p} - \frac{c_{p} - c_{i}}{e^{x} - 1}} \right)} + J_{v} - \frac{{J_{v}\left( {e^{x} - 1} \right)} - {\left( {c_{p} - c_{i}} \right)e^{x}\frac{J_{v}\left( {1 - \sigma} \right)}{PS}\frac{\partial J_{v}}{\partial c_{p}}}}{\left( {e^{x} - 1} \right)^{2}}} \right)}} \\{= {\left( {1 - \sigma} \right)\left( {{\frac{\partial J_{v}}{\partial c_{p}}\left( {c_{p} - \frac{c_{p} - c_{i}}{e^{x} - 1}} \right)} + J_{v} - \frac{{J_{v}\left( {e^{x} - 1} \right)} - {\left( {c_{p} - c_{i}} \right){xe}^{x}\frac{\partial J_{v}}{\partial c_{p}}}}{\left( {e^{x} - 1} \right)^{2}}} \right)}} \\{{= {\left( {1 - \sigma} \right)\left( {{\left( {c_{p} - \frac{c_{p} - c_{i}}{e^{x} - 1} + \frac{\left( {c_{p} - c_{i}} \right){xe}^{x}}{\left( {e^{x} - 1} \right)^{2}}} \right)\frac{\partial J_{v}}{\partial c_{p}}} + J_{v} - \frac{J_{v}}{e^{x} - 1}} \right)}},} \\{\frac{\partial J_{s}}{\partial c_{p}} = {\left( {1 - \sigma} \right){\left( {{\left( {c_{p} - \frac{c_{p} - c_{i}}{e^{x} - 1} + \frac{\left( {c_{p} - c_{i}} \right){xe}^{x}}{\left( {e^{x} - 1} \right)^{2}}} \right)\frac{\partial J_{v}}{\partial c_{p}}} + {J_{v}\left( {1 - \frac{1}{e^{x} - 1}} \right)}} \right).}}}\end{matrix}$

Hence,

$\begin{matrix}{\frac{\partial J_{s}}{\partial c_{p}} = \left\{ \begin{matrix}{\left( {1 - \sigma} \right)\left( {{\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1} + \frac{\left( {c_{p} - c_{i}} \right){xe}^{x}}{\left( {e^{x} - 1} \right)^{2}}} \right)\frac{\partial J_{v}}{\partial c_{p}}} - \frac{J_{v}}{e^{x} - 1}} \right)} & {{{{if}\mspace{14mu} J_{v}} > 0},} \\0 & {{{{if}\mspace{14mu} J_{v}} = 0},} \\{\left( {1 - \sigma} \right)\left( {{\left( {c_{p} - \frac{c_{p} - c_{i}}{e^{x} - 1} + \frac{\left( {c_{p} - c_{i}} \right){xe}^{x}}{\left( {e^{x} - 1} \right)^{2}}} \right)\frac{\partial J_{v}}{\partial c_{p}}} + {J_{v}\left( {1 - \frac{1}{e^{x} - 1}} \right)}} \right)} & {{{if}\mspace{14mu} J_{v}} < 0.}\end{matrix} \right.} & (20)\end{matrix}$

Assuming J_(v)>0, the partial derivative of J_(s) with respect to c_(i)is

$\begin{matrix}{\frac{\partial J_{s}}{\partial c_{i}} = {\left( {1 - \sigma} \right)\left( {{\frac{\partial J_{v}}{\partial c_{i}}\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1}} \right)} + {J_{v}\left( {1 - \frac{{\left( {e^{x} - 1} \right)\left( {- 1} \right)} - {\left( {c_{p} - c_{i}} \right)e^{x}\frac{1 - \sigma}{PS}\frac{\partial J_{v}}{\partial c_{i}}}}{\left( {e^{x} - 1} \right)^{2}}} \right)}} \right)}} \\{= {\left( {1 - \sigma} \right)\left( {{\frac{\partial J_{v}}{\partial c_{i}}\left( {c_{p} - \frac{c_{p} - c_{i}}{e^{x} - 1}} \right)} + J_{v} - \frac{{J_{v}\left( {e^{x} - 1} \right)} - {\left( {c_{p} - c_{i}} \right)e^{x}\frac{J_{v}\left( {1 - \sigma} \right)}{PS}\frac{\partial J_{v}}{\partial c_{i}}}}{\left( {e^{x} - 1} \right)^{2}}} \right)}} \\{= {\left( {1 - \sigma} \right)\left( {{\frac{\partial J_{v}}{\partial c_{i}}\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1}} \right)} + J_{v} - \frac{{J_{v}\left( {e^{x} - 1} \right)} - {\left( {c_{p} - c_{i}} \right){xe}^{x}\frac{\partial J_{v}}{\partial c_{i}}}}{\left( {e^{x} - 1} \right)^{2}}} \right)}} \\{{= {\left( {1 - \sigma} \right)\left( {{\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1} + \frac{\left( {c_{p} - c_{i}} \right){xe}^{x}}{\left( {e^{x} - 1} \right)^{2}}} \right)\frac{\partial J_{v}}{\partial c_{i}}} + J_{v} - \frac{J_{v}}{e^{x} - 1}} \right)}},} \\{\frac{\partial J_{s}}{\partial c_{i}} = {\left( {1 - \sigma} \right){\left( {{\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1} + \frac{\left( {c_{p} - c_{i}} \right){xe}^{x}}{\left( {e^{x} - 1} \right)^{2}}} \right)\frac{\partial J_{v}}{\partial c_{i}}} + {J_{v}\left( {1 - \frac{1}{e^{x} - 1}} \right)}} \right).}}}\end{matrix}$

When J_(v)<0, we obtain

$\begin{matrix}{\frac{\partial J_{s}}{\partial c_{i}} = {\left( {1 - \sigma} \right)\left( {{\frac{\partial J_{v}}{\partial c_{p}}\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1}} \right)} + {J_{v}\left( {- \frac{\left( {e^{x} - 1} \right) - {\left( {c_{p} - c_{i}} \right)e^{x}\frac{1 - \sigma}{PS}\frac{\partial J_{v}}{\partial c_{i}}}}{\left( {e^{x} - 1} \right)^{2}}} \right)}} \right)}} \\{= {\left( {1 - \sigma} \right)\left( {{\frac{\partial J_{v}}{\partial c_{i}}\left( {c_{p} - \frac{c_{p} - c_{i}}{e^{x} - 1}} \right)} + J_{v} - \frac{{J_{v}\left( {e^{x} - 1} \right)} - {\left( {c_{p} - c_{i}} \right)e^{x}\frac{J_{v}\left( {1 - \sigma} \right)}{PS}\frac{\partial J_{v}}{\partial c_{i}}}}{\left( {e^{x} - 1} \right)^{2}}} \right)}} \\{= {\left( {1 - \sigma} \right)\left( {{\frac{\partial J_{v}}{\partial c_{i}}\left( {c_{p} - \frac{c_{p} - c_{i}}{e^{x} - 1}} \right)} + J_{v} - \frac{{J_{v}\left( {e^{x} - 1} \right)} - {\left( {c_{p} - c_{i}} \right){xe}^{x}\frac{\partial J_{v}}{\partial c_{i}}}}{\left( {e^{x} - 1} \right)^{2}}} \right)}} \\{{\frac{\partial J_{s}}{\partial c_{i}} = {\left( {1 - \sigma} \right)\left( {{\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1} + \frac{\left( {c_{p} - c_{i}} \right){xe}^{x}}{\left( {e^{x} - 1} \right)^{2}}} \right)\frac{\partial J_{v}}{\partial c_{p}}} + J_{v} - \frac{J_{v}}{e^{x} - 1}} \right)}},}\end{matrix}$

Therefore,

$\begin{matrix}{\frac{\partial J_{s}}{\partial c_{p}} = \left\{ \begin{matrix}{\left( {1 - \sigma} \right)\left( {{\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1} + \frac{\left( {c_{p} - c_{i}} \right){xe}^{x}}{\left( {e^{x} - 1} \right)^{2}}} \right)\frac{\partial J_{v}}{\partial c_{i}}} + {J_{v}\left( {1 - \frac{1}{e^{x} - 1}} \right)}} \right)} & {{{{if}\mspace{14mu} J_{v}} > 0},} \\0 & {{{{if}\mspace{14mu} J_{v}} = 0},} \\{\left( {1 - \sigma} \right)\left( {{\left( {c_{p} - \frac{c_{p} - c_{i}}{e^{x} - 1} + \frac{\left( {c_{p} - c_{i}} \right){xe}^{x}}{\left( {e^{x} - 1} \right)^{2}}} \right)\frac{\partial J_{v}}{\partial c_{i}}} - \frac{J_{v}}{e^{x} - 1}} \right)} & {{{if}\mspace{14mu} J_{v}} < 0.}\end{matrix} \right.} & (21)\end{matrix}$

Let F be a column vector whose entries are the right-hand side of ourcapillary model, that is,

$\begin{matrix}{{F = {\begin{pmatrix}F_{1} \\F_{2} \\F_{3} \\F_{4}\end{pmatrix} = \begin{pmatrix}{J_{v} + \kappa - J_{UF}} \\\frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \\{{- J_{v}} - \kappa} \\\frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}}\end{pmatrix}}},} & (22)\end{matrix}$

then the Jacobian of F is given by

$\begin{matrix}{{{Jac}_{F} = \begin{pmatrix}\frac{\partial F_{1}}{\partial V_{p}} & \frac{\partial F_{1}}{\partial c_{p}} & \frac{\partial F_{1}}{\partial V_{1}} & \frac{\partial F_{1}}{\partial c_{i}} \\\frac{\partial F_{2}}{\partial V_{p}} & \frac{\partial F_{2}}{\partial c_{p}} & \frac{\partial F_{2}}{\partial V_{i}} & \frac{\partial F_{2}}{\partial c_{i}} \\\frac{\partial F_{3}}{\partial V_{p}} & \frac{\partial F_{3}}{\partial c_{p}} & \frac{\partial F_{3}}{\partial V_{i}} & \frac{\partial F_{3}}{\partial c_{i}} \\\frac{\partial F_{4}}{\partial V_{p}} & \frac{\partial F_{4}}{\partial c_{p}} & \frac{\partial F_{4}}{\partial V_{i}} & \frac{\partial F_{4}}{\partial c_{i}}\end{pmatrix}},{where}} & (23) \\{{\frac{\partial F_{1}}{\partial V_{p}} = 0},{\frac{\partial F_{1}}{\partial c_{p}} = \frac{\partial J_{v}}{\partial c_{p}}},{\frac{\partial F_{1}}{\partial V_{i}} = 0},{\frac{\partial F_{1}}{\partial c_{i}} = \frac{\partial J_{v}}{\partial c_{i}}},{\frac{\partial F_{2}}{\partial V_{p}} = {{- \frac{1}{V_{p}^{2}}}\left( {J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}} \right)}},{\frac{\partial F_{2}}{\partial c_{p}} = {\frac{1}{V_{p}}\left( {\frac{\partial J_{s}}{\partial c_{p}} - {c_{p}\frac{\partial J_{v}}{\partial c_{p}}} - \left( {J_{v} + \kappa - J_{UF}} \right)} \right)}},{\frac{\partial F_{2}}{\partial V_{i}} = 0},{\frac{\partial F_{2}}{\partial c_{i}} = {\frac{1}{V_{P}}\left( {\frac{\partial J_{s}}{\partial c_{i}} - {c_{p}\frac{\partial J_{v}}{\partial c_{i}}}} \right)}},{\frac{\partial F_{3}}{\partial V_{p}} = 0},{\frac{\partial F_{3}}{\partial c_{p}} = {- \frac{\partial J_{v}}{\partial c_{p}}}},{\frac{\partial F_{3}}{\partial V_{i}} = 0},{\frac{\partial F_{3}}{\partial c_{i}} = {- \frac{\partial J_{v}}{\partial c_{i}}}},{\frac{\partial F_{4}}{\partial V_{p}} = 0},{\frac{\partial F_{4}}{\partial c_{p}} = {\frac{1}{V_{i}}\left( {{- \frac{\partial J_{s}}{\partial c_{p}}} + {c_{i}\frac{\partial J_{v}}{\partial c_{p}}}} \right)}},{\frac{\partial F_{4}}{\partial V_{i}} = {{- \frac{1}{V_{i}^{2}}}\left( {{- J_{s}} + {c_{i}\left( {J - v + \kappa} \right)}} \right)}},{\frac{\partial F_{4}}{\partial c_{p}} = {\frac{1}{V_{i}}{\left( {{- \frac{\partial J_{s}}{\partial c_{p}}} + {c_{i}\frac{\partial J_{v}}{\partial c_{p}}} + \left( {J_{v} + \kappa} \right)} \right).}}}} & \;\end{matrix}$

If we set u=(V_(p), c_(p), V_(i), c_(i))^(T) and θ=(L_(p), σ, PS, P_(c),P_(i), α_(p) ₁ , α_(p) ₂ , α_(i) ₁ , α_(i) ₂ , κ, α, J_(UF)), thesensitivity equations with respect to a certain parameter θ_(i)(assuming continuity conditions are satisfied) can be written as

$\begin{matrix}{{\frac{d}{dt}\frac{\partial u}{\partial\theta_{i}}} = {{{\frac{\partial F}{\partial u}\frac{\partial u}{\partial\theta_{i}}} + \frac{\partial F}{\partial\theta_{i}}} = {{{Jac}_{F}\frac{\partial u}{\partial\theta_{i}}} + {\frac{\partial F}{\partial\theta_{i}}.}}}} & (24)\end{matrix}$

Sensitivity with Respect to L_(p)

To derive the sensitivity with respect to L_(p), we need

$\begin{matrix}{\frac{\partial J_{v}}{\partial L_{p}} = {{\sigma \left( {\pi_{p} - \pi_{i}} \right)} - {\left( {P_{c} - P_{i}} \right).}}} & (25)\end{matrix}$

It clearly follows that

$\begin{matrix}{{\frac{\partial}{\partial L_{p}}\left( {e^{x} - 1} \right)} = {e^{x}\frac{1 - \sigma}{PS}{\frac{\partial J_{v}}{\partial L_{p}}.}}} & (26)\end{matrix}$

For J_(v)>0, the partial derivative of J_(s) with respect to L_(p) isderived as

$\begin{matrix}\begin{matrix}{\frac{\partial J_{s}}{\partial L_{p}} = {\left( {1 - \sigma} \right)\left( {{\frac{\partial J_{v}}{\partial L_{p}}\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1}} \right)} + {J_{v}\left( \frac{\left( {c_{p} - c_{i}} \right)e^{x}\frac{1 - \sigma}{PS}\frac{\partial J_{v}}{\partial L_{p}}}{\left( {e^{x} - 1} \right)^{2}} \right)}} \right)}} \\{{= {\left( {1 - \sigma} \right)\left( {\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1}} \right) + {\frac{J_{v}\left( {1 - \sigma} \right)}{PS}\left( \frac{\left( {c_{p} - c_{i}} \right)e^{x}}{\left( {e^{x} - 1} \right)^{2}} \right)}} \right)\frac{\partial J_{v}}{\partial L_{p}}}},} \\{\frac{\partial J_{s}}{\partial L_{p}} = {\left( {1 - \sigma} \right)\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1} + \frac{\left( {c_{p} - c_{i}} \right){xe}^{x}}{\left( {e^{x} - 1} \right)^{2}}} \right){\frac{\partial J_{v}}{\partial L_{p}}.}}}\end{matrix} & \;\end{matrix}$

Similar computation applies for J_(v)<0 and therefore we obtain

$\begin{matrix}{\frac{\partial J_{s}}{\partial L_{p}} = \left\{ \begin{matrix}{\left( {1 - \sigma} \right)\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1} + \frac{\left( {c_{p} - c_{i}} \right){xe}^{x}}{\left( {e^{x} - 1} \right)^{2}}} \right)\frac{\partial\; J_{v}}{\partial L_{p}}} & {{{{{if}\mspace{14mu} J_{v}} > 0},}\mspace{14mu}} \\0 & {{{{if}\mspace{14mu} J_{v}} = 0},} \\{\left( {1 - \sigma} \right)\left( {c_{p} - \frac{c_{p} - c_{i}}{e^{x} - 1} + \frac{\left( {c_{p} - c_{i}} \right){xe}^{x}}{\left( {e^{x} - 1} \right)^{2}}} \right)\frac{\partial J_{v}}{\partial L_{p}}} & {{{if}\mspace{14mu} J_{v}} < 0.}\end{matrix} \right.} & (27)\end{matrix}$

Now, the sensitivity equations with respect to L_(p) are

${\frac{d}{dt}\begin{pmatrix}\frac{\partial V_{p}}{\partial L_{p}} \\\frac{\partial c_{p}}{\partial L_{p}} \\\frac{\partial V_{i}}{\partial L_{p}} \\\frac{\partial c_{i}}{\partial L_{p}}\end{pmatrix}} = {{{Jac}_{F}\begin{pmatrix}\frac{\partial V_{p}}{\partial L_{p}} \\\frac{\partial c_{p}}{\partial L_{p}} \\\frac{\partial V_{i}}{\partial L_{p}} \\\frac{\partial c_{i}}{\partial L_{p}}\end{pmatrix}} + \begin{pmatrix}\frac{\partial F_{1}}{\partial L_{p}} \\\frac{\partial F_{2}}{\partial L_{p}} \\\frac{\partial F_{3}}{\partial L_{p}} \\\frac{\partial F_{4}}{\partial L_{p}}\end{pmatrix}}$ and  so $\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial V_{p}}{\partial L_{p}} \right)} = {\frac{\partial}{\partial L_{p}}\left( \frac{{dV}_{p}}{dt} \right)}} \\{= {{\frac{\partial}{\partial V_{p}}\left( {J_{v} + \kappa - J_{UF}} \right)\frac{\partial V_{p}}{\partial L_{p}}} + {\frac{\partial}{\partial c_{p}}\left( {J_{v} + \kappa - J_{UF}} \right)\; \frac{\partial c_{p}}{\partial L_{p}}} +}} \\{{{\frac{\partial}{\partial V_{i}}\left( {J_{v} + \kappa - J_{UF}} \right)\frac{\partial V_{i}}{\partial L_{p}}} + {\frac{\partial}{\partial c_{i}}\left( {J_{v} + \kappa - J_{UF}} \right)\frac{\partial c_{i}}{\partial L_{p}}} +}} \\{{\frac{\partial}{\partial L_{p}}\left( {J_{v} + \kappa - J_{UF}} \right)}} \\{{= {{\frac{\partial J_{v}}{\partial c_{p}}\frac{\partial c_{p}}{\partial L_{p}}} + {\frac{\partial J_{v}}{\partial c_{i}}\frac{\partial c_{i}}{\partial L_{p}}} + \frac{\partial J_{v}}{\partial L_{p}}}},}\end{matrix}$ $\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial c_{p}}{\partial L_{p}} \right)} = {\frac{\partial}{\partial L_{p}}\left( \frac{d\; c_{p}}{dt} \right)}} \\{= {{\frac{\partial}{\partial V_{p}}\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial V_{p}}{\partial L_{p}}} +}} \\{{{\frac{\partial}{\partial c_{p}}\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial c_{p}}{\partial L_{p}}} +}} \\{{{\frac{\partial}{\partial V_{i}}\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial V_{i}}{\partial L_{p}}} +}} \\{{{\frac{\partial}{\partial c_{i}}\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial c_{i}}{\partial L_{p}}} +}} \\{{{\frac{\partial}{\partial L_{p}}\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)},}} \\{= {{{- \left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}^{2}} \right)}\frac{\partial V_{p}}{\partial L_{p}}} +}} \\{{{\frac{1}{V_{p}}\left( {\frac{\partial J_{s}}{\partial c_{p}} - {c_{p}\frac{\partial J_{v}}{\partial c_{p}}} - \left( {J_{v} + \kappa - J_{UF}} \right)} \right)\frac{\partial c_{p}}{\partial L_{p}}} +}} \\{{{{\frac{1}{V_{p}}\left( {\frac{\partial J_{s}}{\partial c_{i}} - {c_{p}\frac{\partial J_{v}}{\partial c_{i}}}} \right)\frac{\partial c_{i}}{\partial L_{p}}} + {\frac{1}{V_{p}}\left( {\frac{\partial J_{s}}{\partial L_{p}} - {c_{p}\frac{\partial J_{v}}{\partial L_{p}}}} \right)}},}}\end{matrix}$ $\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial V_{i}}{\partial L_{p}} \right)} = {\frac{\partial}{\partial L_{p}}\left( \frac{{dV}_{i}}{dt} \right)}} \\{= {{{+ \frac{\partial}{\partial V_{p}}}\left( {{- J_{v}} - \kappa} \right)\frac{\partial V_{p}}{\partial}} + {\frac{\partial}{\partial c_{p}}\left( {{- J_{v}} - \kappa} \right)\; \frac{\partial c_{p}}{\partial L_{p}}} +}} \\{{{\frac{\partial}{\partial V_{i}}\left( {{- J_{v}} - \kappa} \right)\frac{\partial V_{i}}{\partial L_{p}}} + {\frac{\partial}{\partial c_{i}}\left( {{- J_{v}} - \kappa} \right)\frac{\partial c_{i}}{\partial L_{p}}} +}} \\{{\frac{\partial}{\partial L_{p}}\left( {{- J_{v}} - \kappa} \right)}} \\{{= {{{- \frac{\partial J_{v}}{\partial c_{p}}}\frac{\partial c_{p}}{\partial L_{p}}} - {\frac{\partial J_{v}}{\partial c_{i}}\frac{\partial c_{i}}{\partial L_{p}}} - \frac{\partial J_{v}}{\partial L_{p}}}},}\end{matrix}$ $\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial c_{i}}{\partial L_{p}} \right)} = {\frac{\partial}{\partial L_{p}}\left( \frac{d\; c_{i}}{dt} \right)}} \\{= {{\frac{\partial}{\partial V_{p}}\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)\frac{\partial V_{p}}{\partial L_{p}}} +}} \\{{{\frac{\partial}{\partial c_{p}}\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)\frac{\partial c_{p}}{\partial L_{p}}} +}} \\{{{\frac{\partial}{\partial V_{i}}\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)\frac{\partial V_{i}}{\partial L_{p}}} +}} \\{{{\frac{\partial}{\partial c_{i}}\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)\frac{\partial c_{i}}{\partial L_{p}}} + {\frac{\partial}{\partial L_{p}}\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)}}} \\{= {{\frac{1}{V_{i}}\left( {{- \frac{\partial J_{s}}{\partial c_{p}}} + {c_{i}\frac{\partial J_{v}}{\partial c_{i}}}} \right)\frac{\partial c_{p}}{\partial L_{p}}} - {\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)\frac{\partial V_{i}}{\partial L_{p}}} +}} \\{{{\frac{1}{V_{i}}\left( {{- \frac{\partial J_{s}}{\partial c_{i}}} + {c_{i}\frac{\partial J_{v}}{\partial c_{i}}} + \left( {J_{v} + \kappa} \right)} \right)\frac{\partial c_{i}}{\partial L_{p}}} +}} \\{{\frac{1}{V_{i}}{\left( {{- \frac{\partial J_{s}}{\partial L_{p}}} + {c_{i}\frac{\partial J_{v}}{\partial L_{p}}}} \right).}}}\end{matrix}$

Sensitivity with Respect to σ

One can easily obtain

$\begin{matrix}{\frac{\partial J_{v}}{\partial\sigma} = {{L_{p}\left( {\pi_{p} - \pi_{i}} \right)}.}} & (28)\end{matrix}$

With

${x = \frac{J_{v}\left( {1 - \sigma} \right)}{PS}},$

it follows that

$\begin{matrix}{{\frac{\partial}{\partial\sigma}\left( {e^{x} - 1} \right)} = {{e^{x}\left( {{\frac{\left( {1 - \sigma} \right)}{PS}\frac{\partial J_{v}}{\partial\sigma}} - \frac{J_{v}}{PS}} \right)}.}} & (29)\end{matrix}$

For J_(v)>0, the partial derivative of J_(s) with respect to σ can beobtained as follows

$\begin{matrix}{\frac{\partial J_{s}}{\partial\sigma} = {{\frac{\partial J_{v}}{\partial\sigma}\left( {1 - \sigma} \right)\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1}} \right)} - {J_{v}\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1}} \right)} +}} \\{{{J_{v}\left( {1 - \sigma} \right)}\left( \frac{\left( {c_{p} - c_{i}} \right){e^{x}\left( {{\frac{\left( {1 - \sigma} \right)}{PS}\; \frac{\partial J_{v}}{\partial\sigma}} - \frac{J_{v}}{PS}} \right)}}{\left( {e^{x} - 1} \right)^{2}} \right)}} \\{= {{\frac{\partial J_{v}}{\partial\sigma}\left( {1 - \sigma} \right)\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1}} \right)} - {J_{v}\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1}} \right)} +}} \\{{\frac{J_{v}\left( {1 - \sigma} \right)}{PS}\left( \frac{\left( {c_{p} - c_{i}} \right){e^{x}\left( {{\left( {1 - \sigma} \right)\frac{\partial J_{v}}{\partial\sigma}} - J_{v}} \right)}}{\left( {e^{x} - 1} \right)^{2}} \right)}} \\{= {{\frac{\partial J_{v}}{\partial\sigma}\left( {1 - \sigma} \right)\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1}} \right)} - {J_{v}\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1}} \right)} +}} \\{{x\left( \frac{\left( {c_{p} - c_{i}} \right){e^{x}\left( {{\left( {1 - \sigma} \right)\; \frac{\partial J_{v}}{\partial\sigma}} - J_{v}} \right)}}{\left( {e^{x} - 1} \right)^{2}} \right)}} \\{= {{\left( {{\left( {1 - \sigma} \right)\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1}} \right)} + \frac{\left( {c_{p} - c_{i}} \right)\left( {1 - \sigma} \right){xe}^{x}}{\left( {e^{x} - 1} \right)^{2}}} \right)\frac{\partial J_{v}}{\partial\sigma}} -}} \\{{{{J_{v}\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1}} \right)} - {J_{v}\left( \frac{\left( {c_{p} - c_{i}} \right){xe}^{x}}{\left( {e^{x} - 1} \right)^{2}} \right)}},}} \\{\frac{\partial J_{s}}{\partial\sigma} = {{\left( {1 - \sigma} \right)\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1} + \frac{\left( {c_{p} - c_{i}} \right){xe}^{x}}{\left( {e^{x} - 1} \right)^{2}}} \right)\frac{\partial J_{v}}{2\; \sigma}} -}} \\{{{J_{v}\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1} + \frac{\left( {c_{p} - c_{i}} \right){xe}^{x}}{\left( {e^{x} - 1} \right)^{2}}} \right)}.}}\end{matrix}$

Slight modifications can be obtained when J_(v)<0. Hence, we have

$\begin{matrix}{\frac{\partial J_{s}}{\partial\sigma} = \left\{ \begin{matrix}\begin{matrix}{{\left( {1 - \sigma} \right)\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1} + \frac{\left( {c_{p} - c_{i}} \right){xe}^{x}}{\left( {e^{x} - 1} \right)^{2}}} \right)\frac{\partial J_{v}}{\partial\sigma}} -} \\{J_{v}\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1} + \frac{\left( {c_{p} - c_{i}} \right){xe}^{x}}{\left( {e^{x} - 1} \right)^{2}}} \right)}\end{matrix} & {{{{if}\mspace{14mu} J_{v}} > 0},} \\0 & {{{{if}\mspace{14mu} J_{v}} = 0},} \\\begin{matrix}{{\left( {1 - \sigma} \right)\left( {c_{p} - \frac{c_{p} - c_{i}}{e^{x} - 1} + \frac{\left( {c_{p} - c_{i}} \right){xe}^{x}}{\left( {e^{x} - 1} \right)^{2}}} \right)\frac{\partial J_{v}}{\partial\sigma}} -} \\{J_{v}\left( {c_{p} - \frac{c_{p} - c_{i}}{e^{x} - 1} + \frac{\left( {c_{p} - c_{i}} \right){xe}^{x}}{\left( {e^{x} - 1} \right)^{2}}} \right)}\end{matrix} & {{{if}\mspace{14mu} J_{v}} < 0.}\end{matrix} \right.} & (30)\end{matrix}$

The sensitivity equations with respect to σ can be obtained as follows

${\frac{d}{dt}\begin{pmatrix}\frac{\partial V_{p}}{\partial\sigma} \\\frac{\partial c_{p}}{\partial\sigma} \\\frac{\partial V_{i}}{\partial\sigma} \\\frac{\partial c_{i}}{\partial\sigma}\end{pmatrix}} = {{{Jac}_{F}\begin{pmatrix}\frac{\partial V_{p}}{\partial\sigma} \\\frac{\partial c_{p}}{\partial\sigma} \\\frac{\partial V_{i}}{\partial\sigma} \\\frac{\partial c_{i}}{\partial\sigma}\end{pmatrix}} + \begin{pmatrix}\frac{\partial F_{1}}{\partial\sigma} \\\frac{\partial F_{2}}{\partial\sigma} \\\frac{\partial F_{3}}{\partial\sigma} \\\frac{\partial F_{4}}{\partial\sigma}\end{pmatrix}}$ and  so $\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial V_{p}}{\partial\sigma} \right)} = {\frac{\partial}{\partial\sigma}\left( \frac{{dV}_{p}}{dt} \right)}} \\{= {{\frac{\partial}{\partial V_{p}}\left( {J_{v} + \kappa - J_{UF}} \right)\frac{\partial V_{p}}{\partial\sigma}} + {\frac{\partial}{\partial c_{p}}\left( {J_{v} + \kappa - J_{UF}} \right)\frac{\partial c_{p}}{\partial\sigma}} +}} \\{{{\frac{\partial}{\partial V_{i}}\left( {J_{v} + \kappa - J_{UF}} \right)\frac{\partial V_{i}}{\partial\sigma}\left( {J_{v} + \kappa - J_{UF}} \right)\frac{\partial c_{i}}{\partial\sigma}} +}} \\{{\frac{\partial}{\partial\sigma}\left( {J_{v} + \kappa - J_{UF}} \right)}} \\{{= {{\frac{\partial J_{v}}{\partial c_{p}}\frac{\partial c_{p}}{\partial\sigma}} + {\frac{\partial J_{v}}{\partial c_{i}}\frac{\partial c_{i}}{\partial\sigma}} + \frac{\partial J_{v}}{\partial\sigma}}},}\end{matrix}$ $\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial c_{p}}{\partial\sigma} \right)} = {\frac{\partial}{\partial\sigma}\left( \frac{d\; c_{p}}{dt} \right)}} \\{= {{\frac{\partial}{\partial V_{p}}\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial V_{p}}{\partial\sigma}} +}} \\{{{\frac{\partial}{\partial c_{p}}\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial c_{p}}{\partial\sigma}} +}} \\{{{\frac{\partial}{\partial V_{i}}\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial V_{i}}{\partial\sigma}} +}} \\{{{\frac{\partial}{\partial c_{i}}\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial c_{i}}{\partial\sigma}} +}} \\{{\frac{\partial}{\partial\sigma}\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)}} \\{= {{{- \left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}^{2}} \right)}\frac{\partial V_{p}}{\partial\sigma}} +}} \\{{{\frac{1}{V_{p}}\left( {\frac{\partial J_{s}}{\partial c_{p}} - {c_{p}\; \frac{\partial J_{v}}{\partial c_{p}}} - \left( {J_{v} + \kappa - J_{UF}} \right)} \right)\frac{\partial c_{p}}{\partial\sigma}} +}} \\{{{{\frac{1}{V_{p}}\left( {\frac{\partial J_{s}}{\partial c_{i}} - {c_{p}\frac{\partial J_{v}}{\partial c_{i}}}} \right)\frac{\partial c_{i}}{\partial\sigma}} + {\frac{1}{V_{p}}\left( {\frac{\partial J_{s}}{\partial\sigma} - {c_{p}\; \frac{\partial J_{v}}{\partial\sigma}}} \right)}},}}\end{matrix}$ $\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial V_{i}}{\partial} \right)} = {\frac{\partial}{\partial\sigma}\left( \frac{{dV}_{i}}{dt} \right)}} \\{= {{\frac{\partial}{\partial V_{p}}\left( {{- J_{v}} - \kappa} \right)\frac{\partial V_{p}}{\partial\sigma}} + {\frac{\partial}{\partial c_{p}}\left( {{- J_{v}} - \kappa} \right)\frac{\partial c_{p}}{\partial\sigma}} +}} \\{{{\frac{\partial}{\partial V_{i}}\left( {{- J_{v}} - \kappa} \right)\frac{\partial V_{i}}{\partial\sigma}} + {\frac{\partial}{\partial c_{i}}\left( {{- J_{v}} - \kappa} \right)\frac{\partial c_{i}}{\partial\sigma}} + {\frac{\partial}{\partial\sigma}\left( {{- J_{v}} - \kappa} \right)}}} \\{{= {{{- \frac{\partial J_{v}}{\partial c_{p}}}\frac{\partial c_{p}}{\partial\sigma}} - {\frac{\partial J_{v}}{\partial c_{i}}\frac{\partial c_{i}}{\partial\sigma}} - \frac{\partial J_{v}}{\partial\sigma}}},}\end{matrix}$ $\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial c_{i}}{\partial\sigma} \right)} = {\frac{\partial}{\partial\sigma}\left( \frac{d\; c_{p}}{dt} \right)}} \\{= {{\frac{\partial}{\partial V_{p}}\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{i}} \right)\frac{\partial V_{p}}{\partial\sigma}} +}} \\{{{\frac{\partial}{\partial c_{p}}\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{i}} \right)\frac{\partial c_{p}}{\partial\sigma}} +}} \\{{{\frac{\partial}{\partial V_{i}}\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{i}} \right)\frac{\partial V_{i}}{\partial\sigma}} +}} \\{{{\frac{\partial}{\partial c_{i}}\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{i}} \right)\frac{\partial c_{i}}{\partial\sigma}} +}} \\{{\frac{\partial}{\partial\sigma}\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{i}} \right)}} \\{= {{\frac{1}{V_{i}}\left( {{- \frac{\partial J_{s}}{\partial c_{i}}} + {c_{i}\frac{\partial J_{v}}{\partial c_{i}}} + \left( {J_{v} + \kappa} \right)} \right)\frac{\partial c_{i}}{\partial\sigma}} + {\frac{1}{V_{i}}{\left( {{- \frac{\partial J_{s}}{\partial\sigma}} + {c_{i}\frac{\partial J_{v}}{\partial\sigma}}} \right).}}}}\end{matrix}$

Sensitivity with Respect to PS

Note that

$\begin{matrix}{\frac{\partial J_{v}}{\partial({PS})} = 0.} & (31)\end{matrix}$

With

${x = \frac{J_{v}\left( {1 - \sigma} \right)}{PS}},$

it follows that

$\begin{matrix}{{\frac{\partial}{\partial({PS})}\left( {e^{x} - 1} \right)} = {{- {e^{x}\left( \frac{J_{v}\left( {1\mspace{14mu} \sigma} \right)}{({PS})^{2}} \right)}} = {\frac{{xe}^{x}}{PS}.}}} & (32)\end{matrix}$

For both cases, J_(v)>0 and J_(v)<0, the partial derivative with respectto PS can be derived as follows

$\frac{\partial J_{s}}{\partial({PS})} = {{{J_{v}\left( {1 - \sigma} \right)}\left( \frac{\left( {c_{p} - c_{i}} \right)\left( \frac{- {xe}^{x}}{PS} \right)}{\left( {e^{x} - 1} \right)^{2}} \right)} = {{\frac{J_{v}\left( {1 - \sigma} \right)}{PS}\left( \frac{\left( {c_{p} - c_{i}} \right)\left( {- {xe}^{x}} \right)}{\left( {e^{x} - 1} \right)^{2}} \right)} = {\frac{\left( {c_{p} - c_{i}} \right)x^{2}e^{x}}{\left( {e^{x} - 1} \right)^{2}}.}}}$

Thus, we have

$\begin{matrix}{\frac{\partial J_{s}}{\partial({PS})} = \left\{ \begin{matrix}{- \frac{\left( {c_{p} - c_{i}} \right)x^{2}e^{x}}{\left( {e^{x} - 1} \right)^{2}}} & {{{{if}\mspace{14mu} J_{v}} \neq 0},} \\0 & {{{if}\mspace{14mu} J_{v}} = 0.}\end{matrix} \right.} & (33)\end{matrix}$

The sensitivity equations with respect to PS can be obtained as follows

${\frac{d}{dt}\begin{pmatrix}\frac{\partial V_{p}}{\partial{PS}} \\\frac{\partial c_{p}}{\partial{PS}} \\\frac{\partial V_{i}}{\partial{PS}} \\\frac{\partial c_{i}}{\partial{PS}}\end{pmatrix}} = {{{Jac}_{F}\begin{pmatrix}\frac{\partial V_{p}}{\partial{PS}} \\\frac{\partial c_{p}}{\partial{PS}} \\\frac{\partial V_{i}}{\partial{PS}} \\\frac{\partial c_{i}}{\partial{PS}}\end{pmatrix}} + \begin{pmatrix}\frac{\partial F_{1}}{\partial{PS}} \\\frac{\partial F_{2}}{\partial{PS}} \\\frac{\partial F_{3}}{\partial{PS}} \\\frac{\partial F_{4}}{\partial{PS}}\end{pmatrix}}$ and  so $\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial V_{p}}{\partial({PS})} \right)} = {\frac{\partial}{\partial({PS})}\left( \frac{{dV}_{p}}{dt} \right)}} \\{= {{\frac{\partial}{\partial V_{p}}\left( {J_{v} + \kappa - J_{UF}} \right)\frac{\partial V_{p}}{\partial\sigma}} + {\frac{\partial}{\partial c_{p}}\left( {J_{v} + \kappa - J_{UF}} \right)\frac{\partial c_{p}}{\partial\sigma}} +}} \\{{{\frac{\partial}{\partial V_{i}}\left( {J_{v} + \kappa - J_{UF}} \right)\frac{\partial V_{i}}{\partial\sigma}\left( {J_{v} + \kappa - J_{UF}} \right)\frac{\partial c_{i}}{\partial\sigma}} +}} \\{{\frac{\partial}{\partial\sigma}\left( {J_{v} + \kappa - J_{UF}} \right)}} \\{{= {{\frac{\partial J_{v}}{\partial c_{p}}\frac{\partial c_{p}}{\partial\sigma}} + {\frac{\partial J_{v}}{\partial c_{i}}\frac{\partial c_{i}}{\partial\sigma}} + \frac{\partial J_{v}}{\partial\sigma}}},}\end{matrix}$ $\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial c_{p}}{\partial({PS})} \right)} = {\frac{\partial}{\partial({PS})}\left( \frac{d\; c_{p}}{dt} \right)}} \\{= {{\frac{\partial}{\partial V_{p}}\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial V_{p}}{\partial({PS})}} +}} \\{{{\frac{\partial}{\partial c_{p}}\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial c_{p}}{\partial({PS})}} +}} \\{{{\frac{\partial}{\partial V_{i}}\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial V_{i}}{\partial({PS})}} +}} \\{{{\frac{\partial}{\partial c_{i}}\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial c_{i}}{\partial({PS})}} +}} \\{{\frac{\partial}{\partial\sigma}\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)}} \\{= {{{- \left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}^{2}} \right)}\frac{\partial V_{p}}{\partial({PS})}} +}} \\{{{\frac{1}{V_{p}}\left( {\frac{\partial J_{s}}{\partial c_{p}} - {c_{p}\; \frac{\partial J_{v}}{\partial c_{p}}} - \left( {J_{v} + \kappa - J_{UF}} \right)} \right)\frac{\partial c_{p}}{\partial({PS})}} +}} \\{{{{\frac{1}{V_{p}}\left( {\frac{\partial J_{s}}{\partial c_{i}} - {c_{p}\frac{\partial J_{v}}{\partial c_{i}}}} \right)\frac{\partial c_{i}}{\partial\sigma}} + {\frac{1}{V_{p}}\left( {\frac{\partial J_{s}}{\partial\sigma} - {c_{p}\; \frac{\partial J_{v}}{\partial({PS})}}} \right)}},}}\end{matrix}$ $\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial V_{i}}{\partial} \right)} = {\frac{\partial}{\partial({PS})}\left( \frac{{dV}_{i}}{dt} \right)}} \\{= {{\frac{\partial}{\partial V_{p}}\left( {{- J_{v}} - \kappa} \right)\frac{\partial V_{p}}{\partial({PS})}} + {\frac{\partial}{\partial c_{p}}\left( {{- J_{v}} - \kappa} \right)\frac{\partial c_{p}}{\partial({PS})}} +}} \\{{{\frac{\partial}{\partial V_{i}}\left( {{- J_{v}} - \kappa} \right)\frac{\partial V_{i}}{\partial({PS})}} + {\frac{\partial}{\partial c_{i}}\left( {{- J_{v}} - \kappa} \right)\frac{\partial c_{i}}{\partial({PS})}} +}} \\{{= {{\frac{\partial}{\partial V_{i}}\left( {{- J_{v}} - \kappa} \right)} - {\frac{\partial J_{v}}{\partial c_{p}}\frac{\partial c_{p}}{\partial\sigma}} - {\frac{\partial J_{v}}{\partial c_{i}}\frac{\partial c_{i}}{\partial\sigma}} - \frac{\partial J_{v}}{\partial\sigma}}},}\end{matrix}$ $\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial c_{p}}{\partial\sigma} \right)} = {\frac{\partial}{\partial({PS})}\left( \frac{d\; c_{p}}{dt} \right)}} \\{= {{\frac{\partial}{\partial V_{p}}\left( \frac{J_{s} - {c_{i}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{i}} \right)\frac{\partial V_{p}}{\partial({PS})}} +}} \\{{{\frac{\partial}{\partial c_{p}}\left( \frac{J_{s} - {c_{i}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{i}} \right)\frac{\partial c_{p}}{\partial({PS})}} +}} \\{{{\frac{\partial}{\partial V_{i}}\left( \frac{J_{s} - {c_{i}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{i}} \right)\frac{\partial V_{i}}{\partial({PS})}} +}} \\{{{\frac{\partial}{\partial c_{i}}\left( \frac{J_{s} - {c_{i}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{i}} \right)\frac{\partial c_{i}}{\partial({PS})}} +}} \\{{\frac{\partial}{\partial({PS})}\left( \frac{J_{s} - {c_{i}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{i}} \right)}} \\{= {{\frac{1}{V_{i}}\left( {{- \frac{\partial J_{s}}{\partial c_{p}}} + {c_{i}\; \frac{\partial J_{v}}{\partial c_{p}}}} \right)\frac{\partial c_{p}}{\partial({PS})}} -}} \\{{{\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + k} \right)}}{V_{i}^{2}} \right)\frac{\partial V_{i}}{\partial({PS})}} +}} \\{{{\frac{1}{V_{i}}\left( {{- \frac{\partial J_{s}}{\partial c_{i}}} + {c_{i}\; \frac{\partial J_{v}}{\partial c_{i}}} + \left( {J_{v} + \kappa} \right)} \right)\frac{\partial c_{i}}{\partial({PS})}} - {\frac{1}{V_{i}}{\frac{\partial J_{s}}{\partial({PS})}.}}}}\end{matrix}$

Sensitivity with Respect to P_(c)

Note that

$\begin{matrix}{\frac{\partial J_{v}}{\partial P_{c}} = {- {L_{p}.}}} & (34)\end{matrix}$

With

${x = \frac{J_{v}\left( {1 - \sigma} \right)}{PS}},$

it follows that

$\begin{matrix}{{\frac{\partial}{\partial P_{c}}\left( {e^{x} - 1} \right)} = {{e^{x}\frac{\left( {1 - \sigma} \right)}{PS}\; \frac{\partial J_{v}}{\partial P_{c}}} = {{- L_{p}}\; \frac{\left( {1 - \sigma} \right)}{PS}{e^{x}.}}}} & (35)\end{matrix}$

For J_(v)>0, the partial derivative of J_(s) with respect to P_(c) is

$\begin{matrix}{\frac{\partial J_{s}}{\partial P_{c}} = {\left( {1 - \sigma} \right)\left( {{\frac{\partial J_{v}}{\partial P_{c}}\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1}} \right)} + {J_{v}\left( \frac{\left( {c_{p} - c_{i}} \right)\left( {{- L_{p}}\; \frac{\left( {1 - \sigma} \right)}{PS}e^{x}} \right)}{\left( {e^{x} - 1} \right)^{2}} \right)}} \right)}} \\{= {\left( {1 - \sigma} \right)\left( {{- {L_{p}\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1}} \right)}} - {L_{p}\left( \frac{\left( {c_{p} - c_{i}} \right)\left( {{- L_{p}}\; \frac{\left( {1 - \sigma} \right)}{PS}e^{x}} \right)}{\left( {e^{x} - 1} \right)^{2}} \right)}} \right)}} \\{{= {\left( {1 - \sigma} \right)\left( {{- {L_{p}\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1}} \right)}} - {L_{p}\left( \frac{\left( {c_{p} - c_{i}} \right){xe}^{x}}{\left( {e^{x} - 1} \right)^{2}} \right)}} \right)}},} \\{\frac{\partial J_{s}}{\partial P_{c}} = {{- {L_{p}\left( {1 - \sigma} \right)}}{\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1} + \frac{\left( {c_{p} - c_{i}} \right){xe}^{x}}{\left( {e^{x} - 1} \right)^{2}}} \right).}}}\end{matrix}$

Similarly,

$\frac{\partial J_{s}}{\partial P_{c}}$

can be derived when J_(v)<0. Thus, we have

$\begin{matrix}{\frac{\partial J_{s}}{\partial P_{c}} = \left\{ \begin{matrix}{{- {L_{p}\left( {1 - \sigma} \right)}}\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1} + \frac{\left( {c_{p} - c_{i}} \right){xe}^{x}}{\left( {e^{x} - 1} \right)^{2}}} \right)} & {{{{if}\mspace{14mu} J_{v}} > 0},} \\0 & {{{{if}\mspace{14mu} J_{v}} = 0},} \\{{- {L_{p}\left( {1 - \sigma} \right)}}\left( {c_{p} - \frac{c_{p} - c_{i}}{e^{x} - 1} + \frac{\left( {c_{p} - c_{i}} \right){xe}^{x}}{\left( {e^{x} - 1} \right)^{2}}} \right)} & {{{if}\mspace{14mu} J_{\,^{v}}} < 0.}\end{matrix} \right.} & (36)\end{matrix}$

The sensitivity equations with respect to P_(c) can be obtained asfollows

${\frac{d}{dt}\begin{pmatrix}\frac{\partial V_{p}}{\partial P_{c}} \\\frac{\partial c_{p}}{\partial P_{c}} \\\frac{\partial V_{i}}{\partial P_{c}} \\\frac{\partial c_{i}}{\partial P_{c}}\end{pmatrix}} = {{{Jac}_{F}\begin{pmatrix}\frac{\partial V_{p}}{\partial P_{c}} \\\frac{\partial c_{p}}{\partial P_{c}} \\\frac{\partial V_{i}}{\partial P_{c}} \\\frac{\partial c_{i}}{\partial P_{c}}\end{pmatrix}} + {\begin{pmatrix}\frac{\partial F_{1}}{\partial P_{c}} \\\frac{\partial F_{2}}{\partial P_{c}} \\\frac{\partial F_{3}}{\partial P_{c}} \\\frac{\partial F_{4}}{\partial P_{c}}\end{pmatrix}\mspace{14mu} {and}\mspace{14mu} {so}}}$$\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial V_{p}}{\partial P_{c}} \right)} = {\frac{\partial}{\partial P_{c}}\left( \frac{{dV}_{p}}{dt} \right)}} \\{= {{\frac{\partial}{\partial V_{p}}\left( {J_{v} + \kappa - J_{UF}} \right)\frac{\partial V_{p}}{\partial P_{c}}} + {\frac{\partial}{\partial c_{p}}\left( {J_{v} + \kappa - J_{UF}} \right)\frac{\partial c_{p}}{\partial P_{c}}} +}} \\{{{\frac{\partial}{\partial V_{i}}\left( {J_{v} + \kappa - J_{UF}} \right)\frac{\partial V_{i}}{\partial P_{c}}} + {\frac{\partial}{\partial c_{i}}\left( {J_{v} + \kappa - J_{UF}} \right)\frac{\partial c_{i}}{\partial P_{c}}} +}} \\{{\frac{\partial}{\partial P_{c}}\left( {J_{v} + \kappa - J_{UF}} \right)}} \\{= {{\frac{\partial J_{v}}{\partial c_{p}}\frac{\partial c_{p}}{\partial P_{c}}} + {\frac{\partial J_{v}}{\partial c_{i}}\frac{\partial c_{i}}{\partial P_{c}}} + \frac{\partial J_{v}}{\partial P_{c}}}} \\{{= {{\frac{\partial J_{v}}{\partial c_{p}}\frac{\partial c_{p}}{\partial P_{c}}} + {\frac{\partial J_{v}}{\partial c_{i}}\frac{\partial c_{i}}{\partial P_{c}}} - L_{p}}},}\end{matrix}$ $\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial c_{p}}{\partial P_{c}} \right)} = {\frac{\partial}{\partial P_{c}}\left( \frac{d\; c_{p}}{dt} \right)}} \\{= {{\frac{\partial}{\partial V_{p}}\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial V_{p}}{\partial P_{c}}} + \frac{\partial}{\partial c_{p}}}} \\{{{\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial c_{p}}{\partial P_{c}}} + \frac{\partial}{\partial V_{i}}}} \\{{{\left( \frac{J_{s} - {c_{p}\left( {J_{v} - \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial V_{i}}{\partial P_{c}}} + \frac{\partial}{\partial c_{i}}}} \\{{{\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial c_{i}}{\partial P_{c}}} + \frac{\partial}{\partial P_{c}}}} \\{\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)} \\{= {{{- \left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}^{2}} \right)}\frac{\partial V_{p}}{\partial P_{c}}} + \frac{1}{V_{p}}}} \\{{{\left( {\frac{\partial J_{s}}{\partial c_{p}} - {c_{p}\frac{\partial J_{v}}{\partial c_{p}}} - \left( {J_{v} + \kappa - J_{UF}} \right)} \right)\frac{\partial c_{p}}{\partial P_{c}}} +}} \\{{{\frac{1}{V_{p}}\left( {\frac{\partial J_{s}}{\partial c_{i}} - {c_{p}\frac{\partial J_{v}}{\partial c_{i}}}} \right)\frac{\partial c_{i}}{\partial P_{c}}} + {\frac{1}{V_{p}}\left( {\frac{\partial J_{s}}{\partial P_{c}} - {c_{p}\frac{\partial J_{v}}{\partial P_{c}}}} \right)}}} \\{= {{{- \left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}^{2}} \right)}\frac{\partial V_{p}}{\partial P_{c}}} + \frac{1}{V_{p}}}} \\{{{\left( {\frac{\partial J_{s}}{\partial c_{p}} - {c_{p}\frac{\partial J_{v}}{\partial c_{p}}} - \left( {J_{v} + \kappa - J_{UF}} \right)} \right)\frac{\partial c_{p}}{\partial P_{c}}} + \frac{1}{V_{p}}}} \\{{{{\left( {\frac{\partial J_{s}}{\partial c_{i}} - {c_{p}\frac{\partial J_{v}}{\partial c_{i}}}} \right)\frac{\partial c_{i}}{\partial P_{c}}} + {\frac{1}{V_{p}}\left( {\frac{\partial J_{s}}{\partial P_{c}} + {L_{p}c_{p}}} \right)}},}}\end{matrix}$ $\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial V_{i}}{\partial P_{c}} \right)} = {\frac{\partial}{\partial P_{c}}\left( \frac{{dV}_{i}}{dt} \right)}} \\{= {{\frac{\partial}{\partial V_{p}}\left( {{- J_{v}} - \kappa} \right)\frac{\partial V_{p}}{\partial P_{c}}} + {\frac{\partial}{\partial c_{p}}\left( {{- J_{v}} - \kappa} \right)\frac{\partial c_{p}}{\partial P_{c}}} + \frac{\partial}{\partial V_{i}}}} \\{{{\left( {{- J_{v}} - \kappa} \right)\frac{\partial V_{i}}{\partial P_{c}}} + {\frac{\partial}{\partial c_{i}}\left( {{- J_{v}} - \kappa} \right)}}} \\{{\frac{\partial c_{i}}{\partial P_{c}} + {\frac{\partial}{\partial P_{c}}\left( {{- J_{v}} - \kappa} \right)}}} \\{= {{{- \frac{\partial J_{v}}{\partial c_{p}}}\frac{\partial c_{p}}{\partial P_{c}}} - {\frac{\partial J_{v}}{\partial c_{i}}\frac{\partial c_{i}}{\partial P_{c}}} - \frac{\partial J_{v}}{\partial P_{c}}}} \\{{= {{{- \frac{\partial J_{v}}{\partial c_{p}}}\frac{\partial c_{p}}{\partial P_{c}}} - {\frac{\partial J_{v}}{\partial c_{i}}\frac{\partial c_{i}}{\partial P_{c}}} + L_{p}}},}\end{matrix}$ $\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial c_{i}}{\partial P_{c}} \right)} = {\frac{\partial}{\partial P_{c}}\left( \frac{d\; c_{i}}{dt} \right)}} \\{= {{\frac{\partial}{\partial V_{p}}\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)\frac{\partial V_{p}}{\partial P_{c}}} + \frac{\partial}{\partial c_{p}}}} \\{{{\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)\frac{\partial c_{p}}{\partial P_{c}}} + \frac{\partial}{\partial V_{i}}}} \\{{{\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)\frac{\partial V_{i}}{\partial P_{c}}} + \frac{\partial}{\partial c_{i}}}} \\{{{\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)\frac{\partial c_{i}}{\partial P_{c}}} + \frac{\partial}{\partial P_{c}}}} \\{\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)} \\{= {{\frac{1}{V_{i}}\left( {{- \frac{\partial J_{s}}{\partial c_{p}}} + {c_{i}\frac{\partial J_{v}}{\partial c_{p}}}} \right)\frac{\partial c_{p}}{\partial P_{c}}} - \left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}^{2}} \right)}} \\{{\frac{\partial V_{i}}{\partial P_{c}} + {\frac{1}{V_{i}}\left( {{- \frac{\partial J_{s}}{\partial c_{i}}} + {c_{i}\frac{\partial J_{v}}{\partial c_{i}}} + \left( {J_{v} + \kappa} \right)} \right)\frac{\partial c_{i}}{\partial P_{c}}} +}} \\{{\frac{1}{V_{i}}\left( {{- \frac{\partial J_{s}}{\partial P_{c}}} + {c_{i}\frac{\partial J_{v}}{\partial P_{c}}}} \right)}} \\{= {{\frac{1}{V_{i}}\left( {{- \frac{\partial J_{s}}{\partial c_{p}}} + {c_{i}\frac{\partial J_{v}}{\partial c_{p}}}} \right)\frac{\partial c_{p}}{\partial P_{c}}} - \left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}^{2}} \right)}} \\{{\frac{\partial V_{i}}{\partial P_{c}} + {\frac{1}{V_{i}}\left( {{- \frac{\partial J_{s}}{\partial c_{i}}} + {c_{i}\frac{\partial J_{v}}{\partial c_{i}}} + \left( {J_{v} + \kappa} \right)} \right)\frac{\partial c_{i}}{\partial P_{c}}} +}} \\{{\frac{1}{V_{i}}{\left( {{- \frac{\partial J_{s}}{\partial P_{c}}} - {L_{p}c_{i}}} \right).}}}\end{matrix}$

Sensitivity with Respect to P_(i)

Note that

$\begin{matrix}{\frac{\partial J_{v}}{\partial P_{i}} = {L_{p}.}} & (37)\end{matrix}$

With

${x = \frac{J_{v}\left( {1 - \sigma} \right)}{PS}},$

it follows that

$\begin{matrix}{{\frac{\partial}{\partial P_{i}}\left( {e^{x} - 1} \right)} = {{e^{x}\frac{\left( {1 - \sigma} \right)}{PS}\frac{\partial J_{v}}{\partial P_{i}}} = {L_{p}\frac{\left( {1 - \sigma} \right)}{PS}{e^{x}.}}}} & (38)\end{matrix}$

For J_(v)>0, the partial derivative of J_(s) with respect to P_(i) is

$\begin{matrix}{\frac{\partial J_{s}}{\partial P_{i}} = {\left( {1 - \sigma} \right)\left( {{\frac{\partial J_{v}}{\partial P_{i}}\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1}} \right)} + {J_{v}\left( \frac{\left( {c_{p} - c_{i}} \right)\left( {L_{p}\frac{\left( {1 - \sigma} \right)}{PS}e^{x}} \right)}{\left( {e^{x} - 1} \right)^{2}} \right)}} \right)}} \\{= {\left( {1 - \sigma} \right)\left( {{L_{p}\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1}} \right)} + {L_{p}\left( \frac{\left( {c_{p} - c_{i}} \right)\left( {\frac{J_{v}\left( {1 - \sigma} \right)}{PS}e^{x}} \right)}{\left( {e^{x} - 1} \right)^{2}} \right)}} \right)}} \\{{= {\left( {1 - \sigma} \right)\left( {{L_{p}\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1}} \right)} + {L_{p}\left( \frac{\left( {c_{p} - c_{i}} \right){xe}^{x}}{\left( {e^{x} - 1} \right)^{2}} \right)}} \right)}},}\end{matrix}$$\mspace{20mu} {\frac{\partial J_{s}}{\partial P_{i}} = {{L_{p}\left( {1 - \sigma} \right)}{\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1} + \frac{\left( {c_{p} - c_{i}} \right){xe}^{x}}{\left( {e^{x} - 1} \right)^{2}}} \right).}}}$

Similarly,

$\frac{\partial J_{s}}{\partial P_{i}}$

can be derived when J_(v)<0. Thus, we have

$\begin{matrix}{\frac{\partial J_{s}}{\partial P_{i}} = \left\{ \begin{matrix}{{L_{p}\left( {1 - \sigma} \right)}\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1} + \frac{\left( {c_{p} - c_{i}} \right){xe}^{x}}{\left( {e^{x} - 1} \right)^{2}}} \right)} & {{{{if}\mspace{14mu} J_{v}} > 0},} \\0 & {{{{if}\mspace{14mu} J_{v}} = 0},} \\{{L_{p}\left( {1 - \sigma} \right)}\left( {c_{p} - \frac{c_{p} - c_{i}}{e^{x} - 1} + \frac{\left( {c_{p} - c_{i}} \right){xe}^{x}}{\left( {e^{x} - 1} \right)^{2}}} \right)} & {{{if}\mspace{14mu} J_{\,^{v}}} < 0.}\end{matrix} \right.} & (39)\end{matrix}$

The sensitivity equations with respect to P_(i) can be obtained asfollows

${\frac{d}{dt}\begin{pmatrix}\frac{\partial V_{p}}{\partial P_{i}} \\\frac{\partial c_{p}}{\partial P_{i}} \\\frac{\partial V_{i}}{\partial P_{i}} \\\frac{\partial c_{i}}{\partial P_{i}}\end{pmatrix}} = {{{Jac}_{F}\begin{pmatrix}\frac{\partial V_{p}}{\partial P_{i}} \\\frac{\partial c_{p}}{\partial P_{i}} \\\frac{\partial V_{i}}{\partial P_{i}} \\\frac{\partial c_{i}}{\partial P_{i}}\end{pmatrix}} + {\begin{pmatrix}\frac{\partial F_{1}}{\partial P_{i}} \\\frac{\partial F_{2}}{\partial P_{i}} \\\frac{\partial F_{3}}{\partial P_{i}} \\\frac{\partial F_{4}}{\partial P_{i}}\end{pmatrix}\mspace{14mu} {and}\mspace{14mu} {so}}}$$\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial V_{p}}{\partial P_{i}} \right)} = {\frac{\partial}{\partial P_{i}}\left( \frac{{dV}_{p}}{dt} \right)}} \\{= {{\frac{\partial}{\partial V_{p}}\left( {J_{v} + \kappa - J_{UF}} \right)\frac{\partial V_{p}}{\partial P_{i}}} + {\frac{\partial}{\partial c_{p}}\left( {J_{v} + \kappa - J_{UF}} \right)}}} \\{{\frac{\partial c_{p}}{\partial P_{i}} + {\frac{\partial}{\partial V_{i}}\left( {J_{v} + \kappa - J_{UF}} \right)\frac{\partial V_{i}}{\partial P_{i}}} + \frac{\partial}{\partial c_{i}}}} \\{{{\left( {J_{v} + \kappa - J_{UF}} \right)\frac{\partial c_{i}}{\partial P_{i}}} + {\frac{\partial}{\partial P_{i}}\left( {J_{v} + \kappa - J_{UF}} \right)}}} \\{= {{\frac{\partial J_{v}}{\partial c_{p}}\frac{\partial c_{p}}{\partial P_{i}}} + {\frac{\partial J_{v}}{\partial c_{i}}\frac{\partial c_{i}}{\partial P_{i}}} + \frac{\partial J_{v}}{\partial P_{i}}}} \\{{= {{\frac{\partial J_{v}}{\partial c_{p}}\frac{\partial c_{p}}{\partial P_{i}}} + {\frac{\partial J_{v}}{\partial c_{i}}\frac{\partial c_{i}}{\partial P_{i}}} + L_{p}}},}\end{matrix}$ $\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial c_{p}}{\partial P_{i}} \right)} = {\frac{\partial}{\partial P_{i}}\left( \frac{d\; c_{p}}{dt} \right)}} \\{= {{\frac{\partial}{\partial V_{p}}\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial V_{p}}{\partial P_{i}}} + \frac{\partial}{\partial c_{p}}}} \\{{{\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial c_{p}}{\partial P_{i}}} + \frac{\partial}{\partial V_{i}}}} \\{{{\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial V_{i}}{\partial P_{i}}} + \frac{\partial}{\partial c_{i}}}} \\{{{\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial c_{i}}{\partial P_{i}}} + \frac{\partial}{\partial P_{i}}}} \\{\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)} \\{= {{{- \left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}^{2}} \right)}\frac{\partial V_{p}}{\partial P_{i}}} + \frac{1}{V_{p}}}} \\{{{\left( {\frac{\partial J_{s}}{\partial c_{p}} - {c_{p}\frac{\partial J_{v}}{\partial c_{p}}} - \left( {J_{v} + \kappa - J_{UF}} \right)} \right)\frac{\partial c_{p}}{\partial P_{i}}} +}} \\{{{\frac{1}{V_{p}}\left( {\frac{\partial J_{s}}{\partial c_{i}} - {c_{p}\frac{\partial J_{v}}{\partial c_{i}}}} \right)\frac{\partial c_{i}}{\partial P_{i}}} + {\frac{1}{V_{p}}\left( {\frac{\partial J_{s}}{\partial P_{i}} - {c_{p}\frac{\partial J_{v}}{\partial P_{i}}}} \right)}}} \\{= {{{- \left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}^{2}} \right)}\frac{\partial V_{p}}{\partial P_{i}}} + \frac{1}{V_{p}}}} \\{{{\left( {\frac{\partial J_{s}}{\partial c_{p}} - {c_{p}\frac{\partial J_{v}}{\partial c_{p}}} - \left( {J_{v} + \kappa - J_{UF}} \right)} \right)\frac{\partial c_{p}}{\partial P_{i}}} + \frac{1}{V_{p}}}} \\{{{{\left( {\frac{\partial J_{s}}{\partial c_{p}} - {c_{p}\frac{\partial J_{v}}{\partial c_{p}}}} \right)\frac{\partial c_{i}}{\partial P_{i}}} + {\frac{1}{V_{p}}\left( {\frac{\partial J_{s}}{\partial P_{i}} - {L_{p}c_{p}}} \right)}},}}\end{matrix}$ $\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial V_{i}}{\partial P_{i}} \right)} = {\frac{\partial}{\partial P_{i}}\left( \frac{{dV}_{i}}{dt} \right)}} \\{= {{\frac{\partial}{\partial V_{p}}\left( {{- J_{v}} - \kappa} \right)\frac{\partial V_{p}}{\partial P_{i}}} + {\frac{\partial}{\partial c_{p}}\left( {{- J_{v}} - \kappa} \right)\frac{\partial c_{p}}{\partial P_{i}}} + \frac{\partial}{\partial V_{i}}}} \\{{{\left( {{- J_{v}} - \kappa} \right)\frac{\partial V_{i}}{\partial P_{i}}} + {\frac{\partial}{\partial c_{i}}\left( {{- J_{v}} - \kappa} \right)\frac{\partial c_{i}}{\partial P_{i}}} + \frac{\partial}{\partial P_{i}}}} \\{\left( {{- J_{v}} - \kappa} \right)} \\{= {{{- \frac{\partial J_{v}}{\partial c_{p}}}\frac{\partial c_{p}}{\partial P_{i}}} - {\frac{\partial J_{v}}{\partial c_{i}}\frac{\partial c_{i}}{\partial P_{i}}} - \frac{\partial J_{v}}{\partial P_{i}}}} \\{{= {{{- \frac{\partial J_{v}}{\partial c_{p}}}\frac{\partial c_{p}}{\partial P_{i}}} - {\frac{\partial J_{v}}{\partial c_{i}}\frac{\partial c_{i}}{\partial P_{i}}} - L_{p}}},}\end{matrix}$ $\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial c_{i}}{\partial P_{i}} \right)} = {\frac{\partial}{\partial P_{i}}\left( \frac{d\; c_{i}}{dt} \right)}} \\{= {{\frac{\partial}{\partial V_{p}}\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)\frac{\partial V_{p}}{\partial P_{i}}} + \frac{\partial}{\partial c_{p}}}} \\{{{\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)\frac{\partial c_{p}}{\partial P_{i}}} + {\frac{\partial}{\partial V_{i}}\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)}}} \\{{\frac{\partial V_{i}}{\partial P_{i}} + {\frac{\partial}{\partial c_{i}}\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)\frac{\partial c_{i}}{\partial P_{i}}} + \frac{\partial}{\partial P_{i}}}} \\{\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)} \\{= {{\frac{1}{V_{i}}\left( {{- \frac{\partial J_{s}}{\partial c_{p}}} + {c_{i}\frac{\partial J_{v}}{\partial c_{p}}}} \right)\frac{\partial c_{p}}{\partial P_{i}}} - {\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}^{2}} \right)\frac{\partial V_{i}}{\partial P_{i}}} +}} \\{{{\frac{1}{V_{i}}\left( {{- \frac{\partial J_{s}}{\partial c_{i}}} + {c_{i}\frac{\partial J_{v}}{\partial c_{i}}} + \left( {J_{v} + \kappa} \right)} \right)\frac{\partial c_{i}}{\partial P_{i}}} + \frac{1}{V_{i}}}} \\{\left( {{- \frac{\partial J_{s}}{\partial P_{i}}} + {c_{i}\frac{\partial J_{v}}{\partial P_{i}}}} \right)} \\{= {{\frac{1}{V_{i}}\left( {{- \frac{\partial J_{s}}{\partial c_{p}}} + {c_{i}\frac{\partial J_{v}}{\partial c_{p}}}} \right)\frac{\partial c_{p}}{\partial P_{i}}} - {\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}^{2}} \right)\frac{\partial V_{i}}{\partial P_{i}}} +}} \\{{{\frac{1}{V_{i}}\left( {{- \frac{\partial J_{s}}{\partial c_{i}}} + {c_{i}\frac{\partial J_{v}}{\partial c_{i}}} + \left( {J_{v} + \kappa} \right)} \right)\frac{\partial c_{i}}{\partial P_{i}}} + \frac{1}{V_{i}}}} \\{{\left( {{- \frac{\partial J_{s}}{\partial P_{i}}} + {L_{p}c_{i}}} \right).}}\end{matrix}$

Sensitivity with Respect to α_(P) ₁

Note that

$\begin{matrix}{\frac{\partial J_{v}}{\partial a_{p_{1}}} = {L_{p}\sigma \; {c_{p}.}}} & (40)\end{matrix}$

With

${x = \frac{J_{v}\left( {1 - \sigma} \right)}{PS}},$

it follows that

$\begin{matrix}{{\frac{\partial}{\partial a_{p_{1}}}\left( {e^{x} - 1} \right)} = {e^{x}\frac{\left( {1 - \sigma} \right)}{PS}{\frac{\partial J_{v}}{\partial a_{p_{1}}}.}}} & (41)\end{matrix}$

For J_(v)>0, the partial derivative of J_(s) with respect to α_(p) ₁ is

$\begin{matrix}{\frac{\partial J_{s}}{\partial a_{p_{1}}} = {\left( {1 - \sigma} \right)\left( {{\frac{\partial J_{v}}{\partial P_{i}}\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1}} \right)} + {J_{v}\left( \frac{\left( {c_{p} - c_{i}} \right)\left( {e^{x}\frac{\left( {1 - \sigma} \right)}{PS}\frac{\partial J_{v}}{\partial a_{p_{1}}}} \right)}{\left( {e^{x} - 1} \right)^{2}} \right)}} \right)}} \\{= {\left( {1 - \sigma} \right)\left( {{\frac{\partial J_{v}}{\partial P_{i}}\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1}} \right)} + {\frac{\partial J_{v}}{\partial P_{i}}\left( \frac{\left( {c_{p} - c_{i}} \right)\left( {\frac{J_{v}\left( {1 - \sigma} \right)}{PS}e^{x}} \right)}{\left( {e^{x} - 1} \right)^{2}} \right)}} \right)}} \\{{= {\left( {1 - \sigma} \right)\left( {\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1}} \right) + \left( \frac{\left( {c_{p} - c_{i}} \right){xe}^{x}}{\left( {e^{x} - 1} \right)^{2}} \right)} \right)\frac{\partial J_{v}}{\partial a_{p_{1}}}}},}\end{matrix}$$\mspace{20mu} {\frac{\partial J_{s}}{\partial P_{i}} = {\left( {1 - \sigma} \right)\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1} + \frac{\left( {c_{p} - c_{i}} \right){xe}^{x}}{\left( {e^{x} - 1} \right)^{2}}} \right){\frac{\partial J_{v}}{\partial a_{p_{1}}}.\mspace{20mu} \frac{\partial J_{s}}{\partial a_{p_{1}}}}}}$

can be derived in a similar manner when J_(v)<0. Thus, we have

$\begin{matrix}{\frac{\partial J_{s}}{\partial a_{p_{1}}} = \left\{ \begin{matrix}{\left( {1 - \sigma} \right)\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1} + \frac{\left( {c_{p} - c_{i}} \right){xe}^{x}}{\left( {e^{x} - 1} \right)^{2}}} \right)\frac{\partial J_{v}}{\partial a_{p_{1}}}} & {{{{if}\mspace{14mu} J_{v}} > 0},} \\0 & {{{{if}\mspace{14mu} J_{v}} = 0},} \\{\left( {1 - \sigma} \right)\left( {c_{p} - \frac{c_{p} - c_{i}}{e^{x} - 1} + \frac{\left( {c_{p} - c_{i}} \right){xe}^{x}}{\left( {e^{x} - 1} \right)^{2}}} \right)\frac{\partial J_{v}}{\partial a_{p_{1}}}} & {{{if}\mspace{14mu} J_{\,^{v}}} < 0.}\end{matrix} \right.} & (39)\end{matrix}$

The sensitivity equations with respect to α_(p) ₁ can be obtained asfollows

${\frac{d}{dt}\begin{pmatrix}\frac{\partial V_{p}}{\partial a_{p_{1}}} \\\frac{\partial c_{p}}{\partial a_{p_{1}}} \\\frac{\partial V_{i}}{\partial a_{p_{1}}} \\\frac{\partial c_{i}}{\partial a_{p_{1}}}\end{pmatrix}} = {{{Jac}_{F}\begin{pmatrix}\frac{\partial V_{p}}{\partial a_{p_{1}}} \\\frac{\partial c_{p}}{\partial a_{p_{1}}} \\\frac{\partial V_{i}}{\partial a_{p_{1}}} \\\frac{\partial c_{i}}{\partial a_{p_{1}}}\end{pmatrix}} + {\begin{pmatrix}\frac{\partial F_{1}}{\partial a_{p_{1}}} \\\frac{\partial F_{2}}{\partial a_{p_{1}}} \\\frac{\partial F_{3}}{\partial a_{p_{1}}} \\\frac{\partial F_{4}}{\partial a_{p_{1}}}\end{pmatrix}\mspace{14mu} {and}\mspace{14mu} {so}}}$$\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial V_{p}}{\partial a_{p_{1}}} \right)} = {\frac{\partial}{\partial a_{p_{1}}}\left( \frac{{dV}_{p}}{dt} \right)}} \\{= {{\frac{\partial}{\partial V_{p}}\left( {J_{v} + \kappa - J_{UF}} \right)\frac{\partial V_{p}}{\partial a_{p_{1}}}} + \frac{\partial}{\partial c_{p}}}} \\{{{\left( {J_{v} + \kappa - J_{UF}} \right)\frac{\partial c_{p}}{\partial a_{p_{1}}}} + \frac{\partial}{\partial V_{i}}}} \\{{{\left( {J_{v} + \kappa - J_{UF}} \right)\frac{\partial V_{i}}{\partial a_{p_{i}}}} + {\frac{\partial}{\partial c_{i}}\left( {J_{v} + \kappa - J_{UF}} \right)\frac{\partial c_{i}}{\partial a_{p_{1}}}} +}} \\{{\frac{\partial}{\partial a_{p_{1}}}\left( {J_{v} + \kappa - J_{UF}} \right)}} \\{{= {{\frac{\partial J_{v}}{\partial c_{p}}\frac{\partial c_{p}}{\partial a_{p_{1}}}} + {\frac{\partial J_{v}}{\partial c_{i}}\frac{\partial c_{i}}{\partial a_{p_{1}}}} + \frac{\partial J_{v}}{\partial a_{p_{1}}}}},}\end{matrix}$ $\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial c_{p}}{\partial a_{p_{1}}} \right)} = {\frac{\partial}{\partial a_{p_{1}}}\left( \frac{d\; c_{p}}{dt} \right)}} \\{= {{\frac{\partial}{\partial V_{p}}\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial V_{p}}{\partial a_{p_{1}}}} + \frac{\partial}{\partial c_{p}}}} \\{{{\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial c_{p}}{\partial a_{p_{1}}}} + \frac{\partial}{\partial V_{i}}}} \\{{{\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial V_{i}}{\partial a_{p_{1}}}} + \frac{\partial}{\partial c_{i}}}} \\{{{\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial c_{i}}{\partial a_{p_{1}}}} + \frac{\partial}{\partial a_{p_{1}}}}} \\{\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)} \\{= {{{- \left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}^{2}} \right)}\frac{\partial V_{p}}{\partial a_{p_{1}}}} + \frac{1}{V_{p}}}} \\{{{\left( {\frac{\partial J_{s}}{\partial c_{p}} - {c_{p}\frac{\partial J_{v}}{\partial c_{p}}} - \left( {J_{v} + \kappa - J_{UF}} \right)} \right)\frac{\partial c_{p}}{\partial a_{p_{1}}}} +}} \\{{{\frac{1}{V_{p}}\left( {\frac{\partial J_{s}}{\partial c_{i}} - {c_{p}\frac{\partial J_{v}}{\partial c_{i}}}} \right)\frac{\partial c_{i}}{\partial a_{p_{1}}}} + \frac{1}{V_{p}}}} \\{{\left( {\frac{\partial J_{s}}{\partial a_{p_{1}}} - {c_{p}\frac{\partial J_{v}}{\partial a_{p_{1}}}}} \right),}}\end{matrix}$ $\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial V_{i}}{\partial a_{p_{1}}} \right)} = {\frac{\partial}{\partial a_{p_{1}}}\left( \frac{{dV}_{i}}{dt} \right)}} \\{= {{\frac{\partial}{\partial V_{p}}\left( {{- J_{v}} - \kappa} \right)\frac{\partial V_{p}}{\partial a_{p_{1}}}} + {\frac{\partial}{\partial c_{p}}\left( {{- J_{v}} - \kappa} \right)\frac{\partial c_{p}}{\partial a_{p_{1}}}} + \frac{\partial}{\partial V_{i}}}} \\{{{\left( {{- J_{v}} - \kappa} \right){\frac{\partial V_{i}}{\partial a_{p_{1}}}++}\frac{\partial}{\partial c_{i}}\left( {{- J_{v}} + \kappa} \right)\frac{\partial c_{i}}{\partial a_{p_{1}}}} + \frac{\partial}{\partial a_{p_{1}}}}} \\{\left( {{- J_{v}} - \kappa} \right)} \\{{= {{{- \frac{\partial J_{v}}{\partial c_{p}}}\frac{\partial c_{p}}{\partial a_{p_{1}}}} - {\frac{\partial J_{v}}{\partial c_{i}}\frac{\partial c_{i}}{\partial a_{p_{1}}}} - \frac{\partial J_{v}}{\partial a_{p_{1}}}}},}\end{matrix}$ $\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial c_{i}}{\partial a_{p_{1}}} \right)} = {\frac{\partial}{\partial a_{p_{1}}}\left( \frac{d\; c_{i}}{dt} \right)}} \\{= {{\frac{\partial}{\partial V_{p}}\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)\frac{\partial V_{p}}{\partial a_{p_{1}}}} + \frac{\partial}{\partial c_{p}}}} \\{{{\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)\frac{\partial c_{p}}{\partial a_{p_{1}}}} + \frac{\partial}{\partial V_{i}}}} \\{{{\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)\frac{\partial V_{i}}{\partial a_{p_{1}}}} + {\frac{\partial}{\partial c_{i}}\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)}}} \\{{\frac{\partial c_{i}}{\partial a_{p_{1}}} + {\frac{\partial}{\partial a_{p_{1}}}\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)}}} \\{= {{\frac{1}{V_{i}}\left( {{- \frac{\partial J_{s}}{\partial c_{p}}} + {c_{i}\frac{\partial J_{v}}{\partial c_{p}}}} \right)\frac{\partial c_{p}}{\partial a_{p_{1}}}} - \left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}^{2}} \right)}} \\{{\frac{\partial V_{i}}{\partial a_{p_{1}}} + {\frac{1}{V_{1}}\left( {{- \frac{\partial J_{s}}{\partial c_{i}}} + {c_{i}\frac{\partial J_{v}}{\partial c_{i}}} + \left( {J_{v} + \kappa} \right)} \right)\frac{\partial c_{i}}{\partial a_{p_{1}}}} +}} \\{{\frac{1}{V_{1}}{\left( {{- \frac{\partial J_{s}}{\partial a_{p_{1}}}} + {c_{i}\frac{\partial J_{v}}{\partial a_{p_{1}}}}} \right).}}}\end{matrix}$

Sensitivity with respect to α_(p) ₂

Note that

$\begin{matrix}{\frac{\partial J_{v}}{\partial a_{p_{2}}} = {L_{p}\sigma \; {c_{p}^{2}.}}} & (43)\end{matrix}$

With

${x = \frac{J_{v}\left( {1 - \sigma} \right)}{PS}},$

it follows that

$\begin{matrix}{{\frac{\partial}{\partial a_{p_{2}}}\left( {e^{x} - 1} \right)} = {e^{x}\frac{\left( {1 - \sigma} \right)}{PS}{\frac{\partial J_{v}}{\partial a_{p_{2}}}.}}} & (44)\end{matrix}$

For J_(v)>0, the partial derivative of J_(s) with respect to α_(p) ₂ is

$\begin{matrix}{\frac{\partial J_{s}}{\partial a_{p_{2}}} = {\left( {1 - \sigma} \right)\left( {{\frac{\partial J_{v}}{\partial a_{p_{2}}}\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1}} \right)} + {J_{v}\left( \frac{\left( {c_{p} - c_{i}} \right)\left( {e^{x}\frac{\left( {1 - \sigma} \right)}{PS}\frac{\partial J_{v}}{\partial a_{p_{2}}}} \right)}{\left( {e^{x} - 1} \right)^{2}} \right)}} \right)}} \\{= {\left( {1 - \sigma} \right)\left( {{\frac{\partial J_{v}}{\partial a_{p_{2}}}\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1}} \right)} + {\frac{\partial J_{v}}{\partial a_{p_{2}}}\left( \frac{\left( {c_{p} - c_{i}} \right)\left( {\frac{J_{v}\left( {1 - \sigma} \right)}{PS}e_{x}} \right)}{\left( {e^{x} - 1} \right)^{2}} \right)}} \right)}} \\{{= {\left( {1 - \sigma} \right)\left( {\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1}} \right) + \left( \frac{\left( {c_{p} - c_{i}} \right){xe}^{x}}{\left( {e^{x} - 1} \right)^{2}} \right)} \right)\frac{\partial J_{v}}{\partial a_{p_{2}}}}},}\end{matrix}$$\mspace{20mu} {\frac{\partial J_{s}}{\partial a_{p_{2}}} = {\left( {1 - \sigma} \right)\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1} + \frac{\left( {c_{p} - c_{i}} \right){xe}^{x}}{\left( {e^{x} - 1} \right)^{2}}} \right){\frac{\partial J_{v}}{\partial a_{p_{2}}}.\mspace{20mu} \frac{\partial J_{s}}{\partial a_{p_{2}}}}}}$

can be derived in a similar manner when J_(v)<0. Thus, we have

$\begin{matrix}{\frac{\partial J_{s}}{\partial a_{p_{2}}} = \left\{ \begin{matrix}{\left( {1 - \sigma} \right)\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1} + \frac{\left( {c_{p} - c_{i}} \right){xe}^{x}}{\left( {e^{x} - 1} \right)^{2}}} \right)\frac{\partial J_{v}}{\partial a_{p_{2}}}} & {{{{if}\mspace{14mu} J_{v}} > 0},} \\0 & {{{{if}\mspace{14mu} J_{v}} = 0},} \\{\left( {1 - \sigma} \right)\left( {c_{p} - \frac{c_{p} - c_{i}}{e^{x} - 1} + \frac{\left( {c_{p} - c_{i}} \right){xe}^{x}}{\left( {e^{x} - 1} \right)^{2}}} \right)\frac{\partial J_{v}}{\partial a_{p_{2}}}} & {{{if}\mspace{14mu} J_{v}} < 0.}\end{matrix} \right.} & (45)\end{matrix}$

The sensitivity equations with respect to α_(p) ₂ can be obtained asfollows

${\frac{d}{dt}\begin{pmatrix}\frac{\partial V_{p}}{\partial a_{p_{2}}} \\\frac{\partial c_{p}}{\partial a_{p_{2}}} \\\frac{\partial V_{i}}{\partial a_{p_{2}}} \\\frac{\partial c_{i}}{\partial a_{p_{2}}}\end{pmatrix}} = {{{Jac}_{F}\begin{pmatrix}\frac{\partial V_{p}}{\partial a_{p_{2}}} \\\frac{\partial c_{p}}{\partial a_{p_{2}}} \\\frac{\partial V_{i}}{\partial a_{p_{2}}} \\\frac{\partial c_{i}}{\partial a_{p_{2}}}\end{pmatrix}} + {\begin{pmatrix}\frac{\partial F_{1}}{\partial a_{p_{2}}} \\\frac{\partial F_{2}}{\partial a_{p_{2}}} \\\frac{\partial F_{3}}{\partial a_{p_{2}}} \\\frac{\partial F_{4}}{\partial a_{p_{2}}}\end{pmatrix}\mspace{14mu} {and}\mspace{14mu} {so}}}$$\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial V_{p}}{\partial a_{p_{2}}} \right)} = {\frac{\partial}{\partial a_{p_{2}}}\left( \frac{{dV}_{p}}{dt} \right)}} \\{= {{\frac{\partial}{\partial V_{p}}\left( {J_{v} + \kappa - J_{UF}} \right)\frac{\partial V_{p}}{\partial a_{p_{2}}}} + {\frac{\partial}{\partial c_{p}}\left( {J_{v} + \kappa - J_{UF}} \right)\frac{\partial c_{p}}{\partial a_{p_{2}}}} +}} \\{{{\frac{\partial}{\partial V_{i}}\left( {J_{v} + \kappa - J_{UF}} \right)\frac{\partial V_{i}}{\partial a_{p_{2}}}} + {\frac{\partial}{\partial c_{i}}\left( {J_{v} + \kappa - J_{UF}} \right)\frac{\partial c_{i}}{\partial a_{p_{2}}}} +}} \\{{\frac{\partial}{\partial a_{p_{2}}}\left( {J_{v} + \kappa - J_{UF}} \right)}} \\{{= {{\frac{\partial J_{v}}{\partial c_{p}}\frac{\partial c_{p}}{\partial a_{p_{2}}}} + {\frac{\partial J_{v}}{\partial c_{i}}\frac{\partial c_{i}}{\partial a_{p_{2}}}} + \frac{\partial J_{v}}{\partial a_{p_{2}}}}},}\end{matrix}$ $\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial c_{p}}{\partial a_{p_{2}}} \right)} = {\frac{\partial}{\partial a_{p_{2}}}\left( \frac{d\; c_{p}}{dt} \right)}} \\{= {{\frac{\partial}{\partial V_{p}}\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial V_{p}}{\partial a_{p_{2}}}} +}} \\{{{\frac{\partial}{\partial c_{p}}\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial c_{p}}{\partial a_{p_{2}}}} + \frac{\partial}{\partial V_{i}}}} \\{{{\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial V_{i}}{\partial a_{p_{2}}}} + \frac{\partial}{\partial c_{i}}}} \\{{{\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial c_{i}}{\partial a_{p_{2}}}} + \frac{\partial}{\partial a_{p_{2}}}}} \\{\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)} \\{= {{{- \left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}^{2}} \right)}\frac{\partial V_{p}}{\partial a_{p_{2}}}} + \frac{1}{V_{p}}}} \\{{{\left( {\frac{\partial J_{s}}{\partial c_{p}} - {c_{p}\frac{\partial J_{v}}{\partial c_{p}}} - \left( {J_{v} + \kappa - J_{UF}} \right)} \right)\frac{\partial c_{p}}{\partial a_{p_{2}}}} +}} \\{{{{\frac{1}{V_{p}}\left( {\frac{\partial J_{s}}{\partial c_{i}} - {c_{p}\frac{\partial J_{v}}{\partial c_{i}}}} \right)\frac{\partial c_{i}}{\partial a_{p_{2}}}} + {\frac{1}{V_{p}}\left( {\frac{\partial J_{s}}{\partial a_{p_{2}}} - {c_{p}\frac{\partial J_{v}}{\partial a_{p_{2}}}}} \right)}},}}\end{matrix}$ $\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial V_{i}}{\partial a_{p_{2}}} \right)} = {\frac{\partial}{\partial a_{p_{2}}}\left( \frac{{dV}_{i}}{dt} \right)}} \\{= {{{+ \frac{\partial}{\partial V_{p}}}\left( {{- J_{v}} - \kappa} \right)\frac{\partial V_{p}}{\partial a_{p_{2}}}} + {\frac{\partial}{\partial c_{p}}\left( {{- J_{v}} - \kappa} \right)\frac{\partial c_{p}}{\partial a_{p_{2}}}} +}} \\{{{\frac{\partial}{\partial V_{i}}\left( {{- J_{v}} - \kappa} \right)\frac{\partial V_{i}}{\partial a_{p_{2}}}} + {\frac{\partial}{\partial c_{i}}\left( {{- J_{v}} + \kappa} \right)\frac{\partial c_{i}}{\partial a_{p_{2}}}} + {\frac{\partial}{\partial a_{p_{2}}}\left( {{- J_{v}} - \kappa} \right)}}} \\{{= {{{- \frac{\partial J_{v}}{\partial c_{p}}}\frac{\partial c_{p}}{\partial a_{p_{2}}}} - {\frac{\partial J_{v}}{\partial c_{i}}\frac{\partial c_{i}}{\partial a_{p_{2}}}} - \frac{\partial J_{v}}{\partial a_{p_{2}}}}},}\end{matrix}$ $\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial c_{i}}{\partial a_{p_{2}}} \right)} = {\frac{\partial}{\partial a_{p_{2}}}\left( \frac{d\; c_{i}}{dt} \right)}} \\{= {{\frac{\partial}{\partial V_{p}}\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)\frac{\partial V_{p}}{\partial a_{p_{2}}}} + \frac{\partial}{\partial c_{p}}}} \\{{{\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)\frac{\partial c_{p}}{\partial a_{p_{2}}}} + {\frac{\partial}{\partial V_{i}}\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)\frac{\partial V_{i}}{\partial a_{p_{2}}}} +}} \\{{{\frac{\partial}{\partial c_{i}}\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)\frac{\partial c_{i}}{\partial a_{p_{2}}}} + {\frac{\partial}{\partial a_{p_{2}}}\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)}}} \\{= {{\frac{1}{V_{i}}\left( {{- \frac{\partial J_{s}}{\partial c_{p}}} + {c_{i}\frac{\partial J_{v}}{\partial c_{p}}}} \right)\frac{\partial c_{p}}{\partial a_{p_{2}}}} - {\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}^{2}} \right)\frac{\partial V_{i}}{\partial a_{p_{2}}}} +}} \\{{{\frac{1}{V_{i}}\left( {{- \frac{\partial J_{s}}{\partial c_{i}}} + {c_{i}\frac{\partial J_{v}}{\partial c_{i}}} + \left( {J_{v} + \kappa} \right)} \right)\frac{\partial c_{i}}{\partial a_{p_{2}}}} + \frac{1}{V_{i}}}} \\{{\left( {{- \frac{\partial J_{s}}{\partial a_{p_{1}}}} + {c_{i}\frac{\partial J_{v}}{\partial a_{p_{2}}}}} \right).}}\end{matrix}$

Sensitivity with Respect to a_(i) ₁

Note that

$\begin{matrix}{\frac{\partial J_{v}}{\partial a_{i_{1}}} = {{- L_{p}}\sigma \; {c_{i}.}}} & (46)\end{matrix}$

With

${x = \frac{J_{v}\left( {1 - \sigma} \right)}{PS}},$

it follows that

$\begin{matrix}{{\frac{\partial}{\partial a_{i_{1}}}\left( {e^{x} - 1} \right)} = {e^{x}\frac{\left( {1 - \sigma} \right)}{PS}{\frac{\partial J_{v}}{\partial a_{i_{1}}}.}}} & (47)\end{matrix}$

For J_(v)>0, the partial derivative of J_(s) with respect to α_(i) ₁ is

$\begin{matrix}{\frac{\partial J_{s}}{\partial a_{i_{1}}} = {\left( {1 - \sigma} \right)\left( {{\frac{\partial J_{v}}{\partial a_{i_{1}}}\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1}} \right)} + {J_{v}\left( \frac{\left( {c_{p} - c_{i}} \right)\left( {e^{x}\frac{\left( {1 - \sigma} \right)}{PS}\frac{\partial J_{v}}{\partial a_{i_{1}}}} \right)}{\left( {e^{x} - 1} \right)^{2}} \right)}} \right)}} \\{= {\left( {1 - \sigma} \right)\left( {{\frac{\partial J_{v}}{\partial a_{i_{1}}}\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1}} \right)} + {J_{v}\left( \frac{\left( {c_{p} - c_{i}} \right)\left( {\frac{J_{v}\left( {1 - \sigma} \right)}{PS}e^{x}} \right)}{\left( {e^{x} - 1} \right)^{2}} \right)}} \right)}} \\{{= {\left( {1 - \sigma} \right)\left( {\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1}} \right) + \left( \frac{\left( {c_{p} - c_{i}} \right){xe}^{x}}{\left( {e^{x} - 1} \right)^{2}} \right)} \right)\frac{\partial J_{v}}{\partial a_{i_{1}}}}},}\end{matrix}$$\mspace{20mu} {\frac{\partial J_{s}}{\partial a_{i_{1}}} = {{\left( {1 - \sigma} \right)\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1}} \right)} + {\left( \frac{\left( {c_{p} - c_{i}} \right){xe}^{x}}{\left( {e^{x} - 1} \right)^{2}} \right){\frac{\partial J_{v}}{\partial a_{i_{1}}}.\mspace{20mu} \frac{\partial J_{s}}{\partial a_{i_{1}}}}}}}$

can be derived in a similar manner when J_(v)<0. Thus, we have

$\begin{matrix}{\frac{\partial J_{s}}{\partial a_{p_{2}}} = \left\{ \begin{matrix}{\left( {1 - \sigma} \right)\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1} + \frac{\left( {c_{p} - c_{i}} \right){xe}^{x}}{\left( {e^{x} - 1} \right)^{2}}} \right)\frac{\partial J_{v}}{\partial a_{i_{1}}}} & {{{{if}\mspace{14mu} J_{v}} > 0},} \\0 & {{{{if}\mspace{14mu} J_{v}} = 0},} \\{\left( {1 - \sigma} \right)\left( {c_{p} - \frac{c_{p} - c_{i}}{e^{x} - 1} + \frac{\left( {c_{p} - c_{i}} \right){xe}^{x}}{\left( {e^{x} - 1} \right)^{2}}} \right)\frac{\partial J_{v}}{\partial a_{i_{1}}}} & {{{if}\mspace{14mu} J_{v}} < 0.}\end{matrix} \right.} & (48)\end{matrix}$

The sensitivity equations with respect to α_(i) ₁ can be obtained asfollows

${\frac{d}{dt}\begin{pmatrix}\frac{\partial V_{p}}{\partial a_{i_{1}}} \\\frac{\partial c_{p}}{\partial a_{i_{1}}} \\\frac{\partial V_{i}}{\partial a_{i_{1}}} \\\frac{\partial c_{i}}{\partial a_{i_{1}}}\end{pmatrix}} = {{{Jac}_{F}\begin{pmatrix}\frac{\partial V_{p}}{\partial a_{i_{1}}} \\\frac{\partial c_{p}}{\partial a_{i_{1}}} \\\frac{\partial V_{i}}{\partial a_{i_{1}}} \\\frac{\partial c_{i}}{\partial a_{i_{1}}}\end{pmatrix}} + {\begin{pmatrix}\frac{\partial F_{1}}{\partial a_{i_{1}}} \\\frac{\partial F_{2}}{\partial a_{i_{1}}} \\\frac{\partial F_{3}}{\partial a_{i_{1}}} \\\frac{\partial F_{4}}{\partial a_{i_{1}}}\end{pmatrix}\mspace{14mu} {and}\mspace{14mu} {so}}}$$\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial V_{p}}{\partial a_{i_{1}}} \right)} = {\frac{\partial}{\partial a_{i_{1}}}\left( \frac{{dV}_{p}}{dt} \right)}} \\{= {{\frac{\partial}{\partial V_{p}}\left( {J_{v} + \kappa - J_{UF}} \right)\frac{\partial V_{p}}{\partial a_{i_{1}}}} + {\frac{\partial}{\partial c_{p}}\left( {J_{v} + \kappa - J_{UF}} \right)\frac{\partial c_{p}}{\partial a_{i_{1}}}} +}} \\{{{\frac{\partial}{\partial V_{i}}\left( {J_{v} + \kappa - J_{UF}} \right)\frac{\partial V_{i}}{\partial a_{i_{1}}}} + {\frac{\partial}{\partial c_{i}}\left( {J_{v} + \kappa - J_{UF}} \right)\frac{\partial c_{i}}{\partial a_{i_{1}}}} +}} \\{{\frac{\partial}{\partial a_{i_{1}}}\left( {J_{v} + \kappa - J_{UF}} \right)}} \\{{= {{\frac{\partial J_{v}}{\partial c_{p}}\frac{\partial c_{p}}{\partial a_{i_{1}}}} + {\frac{\partial J_{v}}{\partial c_{i}}\frac{\partial c_{i}}{\partial a_{i_{1}}}} + \frac{\partial J_{v}}{\partial a_{i_{1}}}}},}\end{matrix}$ $\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial}{\partial a_{i_{1}}} \right)} = {\frac{\partial}{\partial a_{i_{1}}}\left( \frac{d\; c_{p}}{dt} \right)}} \\{= {{\frac{\partial}{\partial V_{p}}\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial V_{p}}{\partial a_{i_{1}}}} + \frac{\partial}{\partial c_{p}}}} \\{{{\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial c_{p}}{\partial a_{i_{1}}}} + {\frac{\partial}{\partial V_{i}}\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)}}} \\{{\frac{\partial V_{i}}{\partial a_{i_{1}}} + {\frac{\partial}{\partial c_{i}}\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial c_{i}}{\partial a_{i_{1}}}} + \frac{\partial}{\partial a_{i_{1}}}}} \\{\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)} \\{= {{{- \left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}^{2}} \right)}\frac{\partial V_{p}}{\partial a_{i_{1}}}} + \frac{1}{V_{p}}}} \\{{{\left( {\frac{\partial J_{s}}{\partial c_{p}} - {c_{p}\frac{\partial J_{v}}{\partial c_{p}}} - \left( {J_{v} + \kappa - J_{UF}} \right)} \right)\frac{\partial c_{p}}{\partial a_{i_{1}}}} + \frac{1}{V_{p}}}} \\{{{{\left( {\frac{\partial J_{s}}{\partial c_{i}} - {c_{p}\frac{\partial J_{v}}{\partial c_{i}}}} \right)\frac{\partial c_{i}}{\partial a_{i_{1}}}} + {\frac{1}{V_{p}}\left( {\frac{\partial J_{s}}{\partial a_{i_{1}}} - {c_{p}\frac{\partial J_{v}}{\partial a_{i_{1}}}}} \right)}},}}\end{matrix}$ $\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial V_{i}}{\partial a_{i_{1}}} \right)} = {\frac{\partial}{\partial a_{i_{1}}}\left( \frac{{dV}_{i}}{dt} \right)}} \\{= {{\frac{\partial}{\partial V_{p}}\left( {{- J_{v}} - \kappa} \right)\frac{\partial V_{p}}{\partial a_{i_{1}}}} + {\frac{\partial}{\partial c_{p}}\left( {{- J_{v}} - \kappa} \right)\frac{\partial c_{p}}{\partial a_{i_{1}}}} +}} \\{{{\frac{\partial}{\partial V_{i}}\left( {{- J_{v}} - \kappa} \right)\frac{\partial V_{i}}{\partial a_{i_{1}}}} + {\frac{\partial}{\partial c_{i}}\left( {{- J_{v}} - \kappa} \right)\frac{\partial c_{i}}{\partial a_{i_{1}}}} + {\frac{\partial}{\partial a_{i_{1}}}\left( {{- J_{v}} - \kappa} \right)}}} \\{{= {{{- \frac{\partial J_{v}}{\partial c_{p}}}\frac{\partial c_{p}}{\partial a_{i_{1}}}} - {\frac{\partial J_{v}}{\partial c_{i}}\frac{\partial c_{i}}{\partial a_{i_{1}}}} - \frac{\partial J_{v}}{\partial a_{i_{1}}}}},}\end{matrix}$ $\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial c_{i}}{\partial a_{i_{1}}} \right)} = {\frac{\partial}{\partial a_{i_{1}}}\left( \frac{d\; c}{dt} \right)}} \\{= {{\frac{\partial}{\partial V_{p}}\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)\frac{\partial V_{p}}{\partial a_{i_{1}}}} + {\frac{\partial}{\partial c_{p}}\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)}}} \\{{\frac{\partial c_{p}}{\partial a_{i_{1}}} + {\frac{\partial}{\partial V_{i}}\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)\frac{\partial V_{i}}{\partial a_{i_{1}}}} + \frac{\partial}{\partial c_{i}}}} \\{{{\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)\frac{\partial c_{i}}{\partial a_{i_{1}}}} + {\frac{\partial}{\partial a_{i_{2}}}\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)}}} \\{= {{\frac{1}{V_{i}}\left( {{- \frac{\partial J_{s}}{\partial c_{p}}} + {c_{i}\frac{\partial J_{v}}{\partial c_{p}}}} \right)\frac{\partial c_{p}}{\partial a_{i_{1}}}} - {\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}^{2}} \right)\frac{\partial V_{i}}{\partial a_{i_{1}}}} +}} \\{{{\frac{1}{V_{i}}\left( {{- \frac{\partial J_{s}}{\partial c_{i}}} + {c_{i}\frac{\partial J_{v}}{\partial c_{i}}\left( {J_{v} + \kappa} \right)}} \right)\frac{\partial c_{i}}{\partial a_{i_{1}}}} + {\frac{1}{V_{i}}{\left( {{- \frac{\partial J_{s}}{\partial a_{i_{1}}}} + {c_{i}\frac{\partial J_{v}}{\partial a_{i_{1}}}}} \right).}}}}\end{matrix}$

Sensitivity with Respect to α_(i) ₂

Note that

$\begin{matrix}{\frac{\partial J_{v}}{\partial a_{i_{2}}} = {{- L_{p}}\sigma \; {c_{i}^{2}.}}} & (49)\end{matrix}$

With

${x = \frac{J_{v}\left( {1 - \sigma} \right)}{PS}},$

it follows that

$\begin{matrix}{{\frac{\partial}{\partial a_{p_{2}}}\left( {e^{x} - 1} \right)} = {e^{x}\frac{\left( {1 - \sigma} \right)}{PS}{\frac{\partial J_{v}}{\partial a_{i_{2}}}.}}} & (50)\end{matrix}$

For J_(v)>0, the partial derivative of J_(s) with respect to α_(i) ₂ is

$\begin{matrix}{\frac{\partial J_{s}}{\partial a_{i_{2}}} = {\left( {1 - \sigma} \right)\left( {{\frac{\partial J_{v}}{\partial a_{i_{2}}}\left( {c_{1} - \frac{c_{p} - c_{i}}{e^{x} - 1}} \right)} + {J_{v}\left( \frac{\left( {c_{p} - c_{i}} \right)\left( {e^{x}\frac{\left( {1 - \sigma} \right)}{PS}\frac{\partial J_{v}}{\partial a_{i_{2}}}} \right)}{\left( {e^{x} - 1} \right)^{2}} \right)}} \right)}} \\{= {\left( {1 - \sigma} \right)\left( {{\frac{\partial J_{v}}{\partial a_{i_{2}}}\left( {c_{1} - \frac{c_{p} - c_{i}}{e^{x} - 1}} \right)} + {\frac{\partial J_{v}}{\partial a_{i_{2}}}\left( \frac{\left( {c_{p} - c_{i}} \right)\left( {\frac{J_{v}\left( {1 - \sigma} \right)}{PS}e^{x}} \right)}{\left( {e^{x} - 1} \right)^{2}} \right)}} \right)}} \\{{= {\left( {1 - \sigma} \right)\left( {\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1}} \right) + \left( \frac{\left( {c_{p} - c_{i}} \right){xe}^{x}}{\left( {e^{x} - 1} \right)^{2}} \right)} \right)\frac{\partial J_{v}}{\partial a_{i_{2}}}}},}\end{matrix}$$\mspace{20mu} {\frac{\partial J_{v}}{\partial a_{i_{2}}} = {\left( {1 - \sigma} \right)\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1} + \frac{\left( {c_{p} - c_{i}} \right){xe}^{x}}{\left( {e^{x} - 1} \right)^{2}}} \right){\frac{\partial J_{v}}{\partial a_{i_{2}}}.\mspace{20mu} \frac{\partial J_{s}}{\partial a_{i_{2}}}}}}$

can be derived in a similar manner when J_(v)<0. Thus, we have

$\begin{matrix}{\frac{\partial J_{s}}{\partial a_{i_{2}}} = \left\{ \begin{matrix}{\left( {1 - \sigma} \right)\left( {c_{i} - \frac{c_{p} - c_{i}}{e^{x} - 1} + \frac{\left( {c_{p} - c_{i}} \right){xe}^{x}}{\left( {e^{x} - 1} \right)^{2}}} \right)\frac{\partial J_{v}}{\partial a_{i_{2}}}} & {{{{if}\mspace{14mu} J_{v}} > 0},} \\0 & {{{{if}\mspace{14mu} J_{v}} = 0},} \\{\left( {1 - \sigma} \right)\left( {c_{p} - \frac{c_{p} - c_{i}}{e^{x} - 1} + \frac{\left( {c_{p} - c_{i}} \right){xe}^{x}}{\left( {e^{x} - 1} \right)^{2}}} \right)\frac{\partial J_{v}}{\partial a_{i_{2}}}} & {{{if}\mspace{14mu} J_{v}} < 0.}\end{matrix} \right.} & (51)\end{matrix}$

The sensitivity equations with respect to α_(i) ₂ can be obtained asfollows

${\frac{d}{dt}\begin{pmatrix}\frac{\partial V_{p}}{\partial a_{i_{2}}} \\\frac{\partial c_{p}}{\partial a_{i_{2}}} \\\frac{\partial V_{i}}{\partial a_{i_{2}}} \\\frac{\partial c_{i}}{\partial a_{i_{2}}}\end{pmatrix}} = {{{Jac}_{F}\begin{pmatrix}\frac{\partial V_{p}}{\partial a_{i_{2}}} \\\frac{\partial c_{p}}{\partial a_{i_{2}}} \\\frac{\partial V_{i}}{\partial a_{i_{2}}} \\\frac{\partial c_{i}}{\partial a_{i_{2}}}\end{pmatrix}} = {\begin{pmatrix}\frac{\partial F_{1}}{\partial a_{i_{2}}} \\\frac{\partial F_{2}}{\partial a_{i_{2}}} \\\frac{\partial F_{3}}{\partial a_{i_{2}}} \\\frac{\partial F_{4}}{\partial a_{i_{2}}}\end{pmatrix}\mspace{14mu} {and}\mspace{14mu} {so}}}$$\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial V_{p}}{\partial a_{i_{2}}} \right)} = {\frac{\partial}{\partial a_{i_{2}}}\left( \frac{{dV}_{p}}{dt} \right)}} \\{= {{\frac{\partial}{\partial V_{p}}\left( {J_{v} + \kappa - J_{UF}} \right)\frac{\partial V_{p}}{\partial a_{i_{2}}}} + {\frac{\partial}{\partial c_{p}}\left( {J_{v} + \kappa - J_{UF}} \right)}}} \\{{\frac{\partial c_{p}}{\partial a_{i_{2}}} + {\frac{\partial}{\partial V_{i}}\left( {J_{v} + \kappa - J_{UF}} \right)\frac{\partial V_{i}}{\partial a_{i_{2}}}} +}} \\{{{\frac{\partial}{\partial c_{i}}\left( {J_{v} + \kappa - J_{UF}} \right)\frac{\partial c_{i}}{\partial a_{i_{2}}}} + {\frac{\partial}{\partial a_{i_{2}}}\left( {J_{v} + \kappa - J_{UF}} \right)}}} \\{{= {{\frac{\partial J_{v}}{\partial c_{p}}\frac{\partial c_{p}}{\partial a_{i_{2}}}} + {\frac{\partial J_{v}}{\partial c_{i}}\frac{\partial c_{1}}{\partial a_{i_{2}}}} + \frac{\partial J_{v}}{\partial a_{i_{2}}}}},}\end{matrix}$ $\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial c_{p}}{\partial a_{i_{2}}} \right)} = {\frac{\partial}{\partial a_{i_{2}}}\left( \frac{d\; c_{p}}{dt} \right)}} \\{= {{\frac{\partial}{\partial V_{p}}\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial V_{p}}{\partial a_{i_{2}}}} +}} \\{{{\frac{\partial}{\partial c_{p}}\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial c_{p}}{\partial a_{i_{2}}}} +}} \\{{{\frac{\partial}{\partial V_{i}}\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial V_{i}}{\partial a_{i_{2}}}} +}} \\{{{\frac{\partial}{\partial c_{i}}\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial c_{i}}{\partial a_{i_{2}}}} +}} \\{{\frac{\partial}{\partial a_{i_{2}}}\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)}} \\{= {{{- \left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)}\frac{\partial V_{p}}{\partial a_{i_{2}}}} + \frac{1}{V_{p}}}} \\{{{\left( {\frac{\partial J_{s}}{\partial c_{p}} - {c_{p}\frac{\partial J_{v}}{\partial c_{p}}} - \left( {J_{v} + \kappa - J_{UF}} \right)} \right)\frac{\partial c_{p}}{\partial a_{i_{2}}}} +}} \\{{{{\frac{1}{V_{p}}\left( {\frac{\partial J_{s}}{\partial c_{i}} - {c_{p}\frac{\partial J_{v}}{\partial c_{i}}}} \right)\frac{\partial c_{i}}{\partial a_{i_{2}}}} + {\frac{1}{V_{p}}\left( {\frac{\partial J_{s}}{\partial a_{i_{2}}} - {c_{p}\frac{\partial J_{v}}{\partial a_{i_{2}}}}} \right)}},}}\end{matrix}$ $\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial V_{i}}{\partial a_{i_{2}}} \right)} = {\frac{\partial}{\partial a_{i_{2}}}\left( \frac{{dV}_{i}}{dt} \right)}} \\{= {{\frac{\partial}{\partial V_{p}}\left( {{- J_{v}} - \kappa} \right)\frac{\partial V_{p}}{\partial a_{i_{2}}}} + {\frac{\partial}{\partial c_{p}}\left( {{- J_{v}} - \kappa} \right)\frac{\partial c_{p}}{\partial a_{i_{2}}}} +}} \\{{{\frac{\partial}{\partial V_{i}}\left( {{- J_{v}} - \kappa} \right)\frac{\partial V_{i}}{\partial a_{i_{2}}}} + {\frac{\partial}{\partial c_{i}}\left( {{- J_{v}} - \kappa} \right)}}} \\{{\frac{\partial c_{i}}{\partial a_{i_{2}}} + {\frac{\partial}{\partial a_{i_{2}}}\left( {{- J_{v}} - \kappa} \right)}}} \\{{= {{{- \frac{\partial J_{v}}{\partial c_{p}}}\frac{\partial c_{p}}{\partial a_{i_{2}}}} - {\frac{\partial J_{v}}{\partial c_{i}}\frac{\partial c_{i}}{\partial a_{i_{2}}}} - \frac{\partial J_{v}}{\partial a_{i_{2}}}}},}\end{matrix}$ $\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial c_{i}}{\partial a_{i_{2}}} \right)} = {\frac{\partial}{\partial a_{i_{2}}}\left( \frac{d\; c_{i}}{dt} \right)}} \\{= {{{+ \frac{\partial}{\partial V_{p}}}\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)\frac{\partial V_{p}}{\partial a_{i_{2}}}} +}} \\{{{\frac{\partial}{\partial c_{p}}\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)\frac{\partial c_{p}}{\partial a_{i_{2}}}} + \frac{\partial}{\partial V_{i}}}} \\{{{\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)\frac{\partial V_{i}}{\partial a_{i_{2}}}} + \frac{\partial}{\partial c_{i}}}} \\{{\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right){\frac{\partial c_{i}}{\partial a_{i_{2}}}++}\frac{\partial}{\partial a_{i_{2}}}}} \\{\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)} \\{= {{\frac{1}{V_{i}}\left( {\frac{\partial J_{s}}{\partial c_{p}} + {c_{i}\frac{\partial J_{v}}{\partial c_{p}}}} \right)\frac{\partial c_{p}}{\partial a_{i_{2}}}} - {\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)\frac{\partial V_{i}}{\partial a_{i_{2}}}} +}} \\{{{\frac{1}{V_{i}}\left( {{- \frac{\partial J_{s}}{\partial c_{i}}} + {c_{i}\frac{\partial J_{v}}{\partial c_{i}}} + \left( {J_{v} + \kappa} \right)} \right)\frac{\partial c_{i}}{\partial a_{i_{2}}}} + \frac{1}{V_{i}}}} \\{{\left( {{- \frac{\partial J_{s}}{\partial a_{i_{2}}}} + {c_{i}\frac{\partial J_{v}}{\partial a_{i_{2}}}}} \right).}}\end{matrix}$

Sensitivity with Respect to κ

Note that

$\begin{matrix}{\frac{\partial J_{v}}{\partial\kappa} = 0.} & (52)\end{matrix}$

With

${x = \frac{J_{v}\left( {1 - \sigma} \right)}{PS}},$

it follows that

$\begin{matrix}{{{\frac{\partial}{\partial\kappa}\left( {e^{x} - 1} \right)} = 0},} & (53) \\{\frac{\partial J_{s}}{\partial\kappa} = {\alpha.}} & (54)\end{matrix}$

The sensitivity equations with respect to κ can be obtained as follows

${\frac{d}{dt}\begin{pmatrix}\frac{\partial V_{p}}{\partial\kappa} \\\frac{\partial c_{p}}{\partial\kappa} \\\frac{\partial V_{i}}{\partial\kappa} \\\frac{\partial c_{i}}{\partial\kappa}\end{pmatrix}} = {{{Jac}_{F}\begin{pmatrix}\frac{\partial V_{p}}{\partial\kappa} \\\frac{\partial c_{p}}{\partial\kappa} \\\frac{\partial V_{i}}{\partial\kappa} \\\frac{\partial c_{i}}{\partial\kappa}\end{pmatrix}} = {\begin{pmatrix}\frac{\partial F_{1}}{\partial\kappa} \\\frac{\partial F_{2}}{\partial\kappa} \\\frac{\partial F_{3}}{\partial\kappa} \\\frac{\partial F_{4}}{\partial\kappa}\end{pmatrix}\mspace{14mu} {and}\mspace{14mu} {so}}}$$\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial V_{p}}{\partial\kappa} \right)} = {\frac{\partial}{\partial\kappa}\left( \frac{{dV}_{p}}{dt} \right)}} \\{= {{\frac{\partial}{\partial V_{p}}\left( {J_{v} + \kappa - J_{UF}} \right)\frac{\partial V_{p}}{\partial\kappa}} + {\frac{\partial}{\partial c_{p}}\left( {J_{v} + \kappa - J_{UF}} \right)}}} \\{{\frac{\partial c_{p}}{\partial\kappa} + {\frac{\partial}{\partial V_{i}}\left( {J_{v} + \kappa - J_{UF}} \right)\frac{\partial V_{i}}{\partial\kappa}} +}} \\{{{\frac{\partial}{\partial c_{i}}\left( {J_{v} + \kappa - J_{UF}} \right)\frac{\partial c_{i}}{\partial\kappa}} + {\frac{\partial}{\partial\kappa}\left( {J_{v} + \kappa - J_{UF}} \right)}}} \\{= {{\frac{\partial J_{v}}{\partial c_{p}}\frac{\partial c_{p}}{\partial\kappa}} + {\frac{\partial J_{v}}{\partial c_{i}}\frac{\partial c_{1}}{\partial\kappa}} + \frac{\partial J_{v}}{\partial\kappa} + 1}} \\{{= {{\frac{\partial J_{v}}{\partial c_{p}}\frac{\partial c_{p}}{\partial\kappa}} + {\frac{\partial J_{v}}{\partial c_{i}}\frac{\partial c_{i}}{\partial\kappa}} + 1}},}\end{matrix}$ $\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial c_{p}}{\partial\kappa} \right)} = {\frac{\partial}{\partial\kappa}\left( \frac{d\; c_{p}}{dt} \right)}} \\{= {{\frac{\partial}{\partial V_{p}}\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial V_{p}}{\partial\kappa}} +}} \\{{{\frac{\partial}{\partial c_{p}}\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial c_{p}}{\partial\kappa}} +}} \\{{{\frac{\partial}{\partial V_{i}}\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial V_{i}}{\partial\kappa}} +}} \\{{{\frac{\partial}{\partial c_{i}}\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial c_{i}}{\partial\kappa}} +}} \\{{\frac{\partial}{\partial\kappa}\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)}} \\{= {{{- \left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)}\frac{\partial V_{p}}{\partial\kappa}} + \frac{1}{V}}} \\{{{\left( {\frac{\partial J_{s}}{\partial c_{p}} - {c_{p}\frac{\partial J_{v}}{\partial c_{p}}} - \left( {J_{v} + \kappa - J_{UF}} \right)} \right)\frac{\partial c_{p}}{\partial\kappa}} +}} \\{{{\frac{1}{V_{p}}\left( {\frac{\partial J_{s}}{\partial c_{i}} - {c_{p}\frac{\partial J_{v}}{\partial c_{i}}}} \right)\frac{\partial c_{i}}{\partial\kappa}} + {\frac{1}{V_{p}}\left( {\frac{\partial J_{s}}{\partial\kappa} - {c_{p}\frac{\partial J_{v}}{\partial\kappa}} - c_{p}} \right)}}} \\{= {{{- \left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}^{2}} \right)}\frac{\partial V_{p}}{\partial\kappa}} + \frac{1}{V}}} \\{{{\left( {\frac{\partial J_{s}}{\partial c_{p}} - {c_{p}\frac{\partial J_{v}}{\partial c_{p}}} - \left( {J_{v} + \kappa - J_{UF}} \right)} \right)\frac{\partial c_{p}}{\partial\kappa}} +}} \\{{{{\frac{1}{V_{p}}\left( {\frac{\partial J_{s}}{\partial c_{i}} - {c_{p}\frac{\partial J_{v}}{\partial c_{i}}}} \right)\frac{\partial c_{i}}{\partial\kappa}} + \frac{\alpha - c_{p}}{V_{p}}},}}\end{matrix}$ $\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial V_{i}}{\partial\kappa} \right)} = {\frac{\partial}{\partial\kappa}\left( \frac{{dV}_{i}}{dt} \right)}} \\{= {{\frac{\partial}{\partial V_{p}}\left( {{- J_{v}} - \kappa} \right)\frac{\partial V_{p}}{\partial\kappa}} + {\frac{\partial}{\partial c_{p}}\left( {{- J_{v}} - \kappa} \right)\frac{\partial c_{p}}{\partial\kappa}} +}} \\{{{\frac{\partial}{\partial V_{i}}\left( {{- J_{v}} - \kappa} \right)\frac{\partial V_{i}}{\partial\kappa}} + {\frac{\partial}{\partial c_{i}}\left( {{- J_{v}} - \kappa} \right)\frac{\partial c_{i}}{\partial\kappa}} + \frac{\partial}{\partial\kappa}}} \\{\left( {{- J_{v}} - \kappa} \right)} \\{{= {{{- \frac{\partial J_{v}}{\partial c_{p}}}\frac{\partial c_{p}}{\partial\kappa}} - {\frac{\partial J_{v}}{\partial c_{i}}\frac{\partial c_{i}}{\partial\kappa}} - 1}},}\end{matrix}$ $\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial c_{i}}{\partial\kappa} \right)} = {\frac{\partial}{\partial\kappa}\left( \frac{d\; c_{i}}{dt} \right)}} \\{= {{\frac{\partial}{\partial V_{p}}\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)\frac{\partial V_{p}}{\partial\kappa}} +}} \\{{{\frac{\partial}{\partial c_{p}}\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)\frac{\partial c_{p}}{\partial\kappa}} +}} \\{{{\frac{\partial}{\partial V_{i}}\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)\frac{\partial V_{i}}{\partial\kappa}} +}} \\{{{\frac{\partial}{\partial c_{i}}\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)\frac{\partial c_{i}}{\partial\kappa}} +}} \\{{\frac{\partial}{\partial\kappa}\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)}} \\{= {{\frac{1}{V_{i}}\left( {{- \frac{\partial J_{s}}{\partial c_{p}}} + {c_{i}\frac{\partial J_{v}}{\partial c_{p}}}} \right)\frac{\partial c_{p}}{\partial\kappa}} - {\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)\frac{\partial V_{i}}{\partial\kappa}} +}} \\{{{\frac{1}{V_{i}}\left( {{- \frac{\partial J_{s}}{\partial c_{i}}} + {c_{i}\frac{\partial J_{v}}{\partial c_{i}}} + \left( {J_{v} + \kappa} \right)} \right)\frac{\partial c_{i}}{\partial\kappa}} + \frac{1}{V_{i}}}} \\{\left( {{- \frac{\partial J_{s}}{\partial\kappa}} + {c_{i}\frac{\partial J_{v}}{\partial\kappa}} + c_{i}} \right)} \\{= {{\frac{1}{V_{i}}\left( {{- \frac{\partial J_{s}}{\partial c_{p}}} + {c_{i}\frac{\partial J_{v}}{\partial c_{p}}}} \right)\frac{\partial c_{p}}{\partial\kappa}} - {\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}^{2}} \right)\frac{\partial V_{i}}{\partial\kappa}} +}} \\{{{\frac{1}{V_{i}}\left( {{- \frac{\partial J_{s}}{\partial c_{i}}} + {c_{i}\frac{\partial J_{v}}{\partial c_{i}}} + \left( {J_{v} + \kappa} \right)} \right)\frac{\partial c_{i}}{\partial\kappa}} + {\frac{{- \alpha} + c_{1}}{V_{i}}.}}}\end{matrix}$

Sensitivity with Respect to α

Note that

$\begin{matrix}{\frac{\partial J_{v}}{\partial\alpha} = 0.} & (55)\end{matrix}$

With

${x = \frac{J_{v}\left( {1 - \sigma} \right)}{PS}},$

it follows that

$\begin{matrix}{{{\frac{\partial}{\partial\alpha}\left( {e^{x} - 1} \right)} = 0},} & (56) \\{\frac{\partial J_{s}}{\partial\alpha} = {\kappa.}} & (57)\end{matrix}$

The sensitivity equations with respect to α can be obtained as follows

${\frac{d}{dt}\begin{pmatrix}\frac{\partial V_{p}}{\partial\alpha} \\\frac{\partial c_{p}}{\partial\alpha} \\\frac{\partial V_{i}}{\partial\alpha} \\\frac{\partial c_{i}}{\partial\alpha}\end{pmatrix}} = {{{Jac}_{F}\begin{pmatrix}\frac{\partial V_{p}}{\partial\alpha} \\\frac{\partial c_{p}}{\partial\alpha} \\\frac{\partial V_{i}}{\partial\alpha} \\\frac{\partial c_{i}}{\partial\alpha}\end{pmatrix}} + {\begin{pmatrix}\frac{\partial F_{1}}{\partial\alpha} \\\frac{\partial F_{2}}{\partial\alpha} \\\frac{\partial F_{3}}{\partial\alpha} \\\frac{\partial F_{4}}{\partial\alpha}\end{pmatrix}\mspace{14mu} {and}\mspace{14mu} {so}}}$$\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial V_{p}}{\partial\alpha} \right)} = {\frac{\partial}{\partial\alpha}\left( \frac{{dV}_{p}}{dt} \right)}} \\{= {{\frac{\partial}{\partial V_{p}}\left( {J_{v} + \kappa - J_{UF}} \right)\frac{\partial V_{p}}{\partial\alpha}} + {\frac{\partial}{\partial c_{p}}\left( {J_{v} + \kappa - J_{UF}} \right)\frac{\partial c_{p}}{\partial\alpha}} +}} \\{{{\frac{\partial}{\partial V_{i}}\left( {J_{v} + \kappa - J_{UF}} \right)\frac{\partial V_{i}}{\partial\alpha}} + {\frac{\partial}{\partial c_{i}}\left( {J_{v} + \kappa - J_{UF}} \right)}}} \\{{\frac{\partial c_{i}}{\partial\alpha} + {\frac{\partial}{\partial\alpha}\left( {J_{v} + \kappa - J_{UF}} \right)}}} \\{= {{\frac{\partial J_{v}}{\partial c_{p}}\frac{\partial c_{p}}{\partial\alpha}} + {\frac{\partial J_{v}}{\partial c_{i}}\frac{\partial c_{i}}{\partial\alpha}} + \frac{\partial J_{v}}{\partial\alpha}}} \\{= {{\frac{\partial J_{v}}{\partial c_{p}}\frac{\partial c_{p}}{\partial\alpha}} + {\frac{\partial J_{v}}{\partial c_{i}}{\frac{\partial c_{i}}{\partial\alpha}.}}}}\end{matrix}$ $\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial c_{p}}{\partial\alpha} \right)} = {\frac{\partial}{\partial\alpha}\left( \frac{d\; c_{p}}{dt} \right)}} \\{= {{\frac{\partial}{\partial V_{p}}\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial V_{p}}{\partial\alpha}} +}} \\{{{\frac{\partial}{\partial c_{p}}\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial c_{p}}{\partial\alpha}} +}} \\{{{\frac{\partial}{\partial V_{i}}\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial V_{i}}{\partial\alpha}} +}} \\{{{\frac{\partial}{\partial c_{i}}\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial c_{i}}{\partial\alpha}} +}} \\{{\frac{\partial}{\partial\alpha}\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)}} \\{= {{{- \left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}^{2}} \right)}\frac{\partial V_{p}}{\partial\alpha}} +}} \\{{{\frac{1}{V_{p}}\left( {\frac{\partial J_{s}}{\partial c_{p}} - {c_{p}\frac{\partial J_{v}}{\partial c_{p}}} - \left( {J_{v} + \kappa - J_{UF}} \right)} \right)\frac{\partial c_{p}}{\partial\alpha}} +}} \\{{{\frac{1}{V_{p}}\left( {\frac{\partial J_{s}}{\partial c_{i}} - {c_{p}\frac{\partial J_{v}}{\partial c_{i}}}} \right)\frac{\partial c_{i}}{\partial\alpha}} + {\frac{1}{V_{p}}\left( {\frac{\partial J_{s}}{\partial\alpha} - {c_{p}\frac{\partial J_{v}}{\partial\alpha}}} \right)}}} \\{= {{{- \left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}^{2}} \right)}\frac{\partial V_{p}}{\partial\alpha}} + \frac{1}{V_{p}}}} \\{{{\left( {\frac{\partial J_{s}}{\partial c_{p}} - {c_{p}\frac{\partial J_{v}}{\partial c_{p}}} - \left( {J_{v} + \kappa - J_{UF}} \right)} \right)\frac{\partial c_{p}}{\partial\alpha}} +}} \\{{{{\frac{1}{V_{p}}\left( {\frac{\partial J_{s}}{\partial c_{i}} - {c_{p}\frac{\partial J_{v}}{\partial c_{i}}}} \right)\frac{\partial c_{i}}{\partial\alpha}} + \frac{\kappa}{V_{p}}},}}\end{matrix}$ $\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial V_{i}}{\partial\alpha} \right)} = {\frac{\partial}{\partial\alpha}\left( \frac{{dV}_{i}}{dt} \right)}} \\{= {{\frac{\partial}{\partial V_{p}}\left( {{- J_{v}} - \kappa} \right)\frac{\partial V_{p}}{\partial\alpha}} + {\frac{\partial}{\partial c_{p}}\left( {{- J_{v}} - \kappa} \right)\frac{\partial c_{p}}{\partial\alpha}} +}} \\{{{\frac{\partial}{\partial V_{i}}\left( {{- J_{v}} - \kappa} \right)\frac{\partial V_{i}}{\partial\alpha}} + {\frac{\partial}{\partial c_{i}}\left( {{- J_{v}} - \kappa} \right)\frac{\partial c_{i}}{\partial\alpha}} + {\frac{\partial}{\partial\alpha}\left( {{- J_{v}} - \kappa} \right)}}} \\{{= {{{- \frac{\partial J_{v}}{\partial c_{p}}}\frac{\partial c_{p}}{\partial\alpha}} - {\frac{\partial J_{v}}{\partial c_{i}}\frac{\partial c_{i}}{\partial\alpha}}}},}\end{matrix}$ $\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial c_{i}}{\partial\alpha} \right)} = {\frac{\partial}{\partial\alpha}\left( \frac{d\; c_{i}}{dt} \right)}} \\{= {{\frac{\partial}{\partial V_{p}}\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)\frac{\partial V_{p}}{\partial\alpha}} + {\frac{\partial}{\partial c_{p}}\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)}}} \\{{\frac{\partial c_{p}}{\partial\alpha} + {\frac{\partial}{\partial V_{i}}\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)\frac{\partial V_{i}}{\partial\alpha}} + \frac{\partial}{\partial c_{i}}}} \\{{{\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)\frac{\partial c_{i}}{\partial\alpha}} + {\frac{\partial}{\partial\alpha}\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)}}} \\{= {{\frac{1}{V_{1}}\left( {{- \frac{\partial J_{s}}{\partial c_{p}}} + {c_{i}\frac{\partial J_{v}}{\partial c_{p}}}} \right)\frac{\partial c_{p}}{\partial\alpha}} - {\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}^{2}} \right)\frac{\partial V_{i}}{\partial\alpha}} +}} \\{{{\frac{1}{V_{i}}\left( {{- \frac{\partial J_{s}}{\partial c_{i}}} + {c_{i}\frac{\partial J_{v}}{\partial c_{i}}} + \left( {J_{v} + \kappa} \right)} \right)\frac{\partial c_{i}}{\partial\alpha}} + {\frac{1}{V_{i}}\left( {{- \frac{\partial J_{s}}{\partial\alpha}} + {c_{i}\frac{\partial J_{v}}{\partial\alpha}}} \right)}}} \\{= {{\frac{1}{V_{i}}\left( {{- \frac{\partial J_{s}}{\partial c_{p}}} + {c_{i}\frac{\partial J_{v}}{\partial c_{p}}}} \right)\frac{\partial c_{p}}{\partial\alpha}} - {\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}^{2}} \right)\frac{\partial V_{i}}{\partial\alpha}} +}} \\{{{\frac{1}{V_{i}}\left( {{- \frac{\partial J_{s}}{\partial c_{i}}} + {c_{i}\frac{\partial J_{v}}{\partial c_{i}}} + \left( {J_{v} + \kappa} \right)} \right)\frac{\partial c_{i}}{\partial\alpha}} - {\frac{\kappa}{V_{i}}.}}}\end{matrix}$

Sensitivity with Respect to J_(UF)

Note that

$\begin{matrix}{\frac{\partial J_{v}}{\partial J_{UF}} = 0.} & (58)\end{matrix}$

With

${x = \frac{J_{v}\left( {1 - \sigma} \right)}{PS}},$

it follows that

$\begin{matrix}{{{\frac{\partial}{\partial J_{UF}}\left( {e^{x} - 1} \right)} = 0},} & (59) \\{\frac{\partial J_{s}}{\partial J_{UF}} = 0.} & (60)\end{matrix}$

The sensitivity equations with respect to J_(UF) can be obtained asfollows

${\frac{d}{dt}\begin{pmatrix}\frac{\partial V_{p}}{\partial J_{UF}} \\\frac{\partial c_{p}}{\partial J_{UF}} \\\frac{\partial V_{i}}{\partial J_{UF}} \\\frac{\partial c_{i}}{\partial J_{UF}}\end{pmatrix}} = {{{Jac}_{F}\begin{pmatrix}\frac{\partial V_{p}}{\partial J_{UF}} \\\frac{\partial c_{p}}{\partial J_{UF}} \\\frac{\partial V_{i}}{\partial J_{UF}} \\\frac{\partial c_{i}}{\partial J_{UF}}\end{pmatrix}} + {\begin{pmatrix}\frac{\partial F_{1}}{\partial J_{UF}} \\\frac{\partial F_{2}}{\partial J_{UF}} \\\frac{\partial F_{3}}{\partial J_{UF}} \\\frac{\partial F_{4}}{\partial J_{UF}}\end{pmatrix}\mspace{14mu} {and}\mspace{14mu} {so}}}$$\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial V_{p}}{\partial J_{UF}} \right)} = {\frac{\partial}{\partial J_{UF}}\left( \frac{{dV}_{p}}{dt} \right)}} \\{= {{\frac{\partial}{\partial V_{p}}\left( {J_{v} - \kappa - J_{UF}} \right)\frac{\partial V_{p}}{\partial J_{UF}}} + \frac{\partial}{\partial c_{p}}}} \\{{{\left( {J_{v} + \kappa - J_{UF}} \right)\frac{\partial c_{p}}{\partial J_{UF}}} + {\frac{\partial}{\partial V_{i}}\left( {J_{v} + \kappa - J_{UF}} \right)\frac{\partial V_{i}}{\partial J_{UF}}} +}} \\{{{\frac{\partial}{\partial c_{i}}\left( {J_{v} + \kappa - J_{UF}} \right)\frac{\partial c_{i}}{\partial J_{UF}}} + {\frac{\partial}{\partial J_{UF}}\left( {J_{v} + \kappa - J_{UF}} \right)}}} \\{{= {{\frac{\partial J_{v}}{\partial c_{p}}\frac{\partial c_{p}}{\partial J_{UF}}} + {\frac{\partial J_{v}}{\partial c_{i}}\frac{\partial c_{i}}{\partial J_{UF}}} - 1}},}\end{matrix}$ $\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial c_{p}}{\partial J_{UF}} \right)} = {\frac{\partial}{\partial J_{UF}}\left( \frac{d\; c_{p}}{dt} \right)}} \\{= {{\frac{\partial}{\partial V_{p}}\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial V_{p}}{\partial J_{UF}}} + \frac{\partial}{\partial c_{p}}}} \\{{{\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial c_{p}}{\partial\alpha}} + \frac{\partial}{\partial V_{i}}}} \\{{{\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial V_{i}}{\partial J_{UF}}} + \frac{\partial}{\partial c_{i}}}} \\{{{\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)\frac{\partial c_{i}}{\partial J_{UF}}} + \frac{\partial}{\partial J_{UF}}}} \\{\left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}} \right)} \\{= {{{- \left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}^{2}} \right)}\frac{\partial V_{p}}{\partial J_{UF}}} + \frac{1}{V_{p}}}} \\{{{\left( {\frac{\partial J_{s}}{\partial c_{p}} - {c_{p}\frac{\partial J_{v}}{\partial c_{p}}} - \left( {J_{v} + \kappa - J_{UF}} \right)} \right)\frac{\partial c_{p}}{\partial J_{UF}}} +}} \\{{{\frac{1}{V_{p}}\left( {\frac{\partial J_{s}}{\partial c_{i}} - {c_{p}\frac{\partial J_{v}}{\partial c_{i}}}} \right)\frac{\partial c_{i}}{\partial J_{UF}}} + \frac{1}{V_{p}}}} \\{\left( {\frac{\partial J_{s}}{\partial J_{UF}} - {c_{p}\frac{\partial J_{v}}{\partial J_{UF}}} + c_{p}} \right)} \\{= {{{- \left( \frac{J_{s} - {c_{p}\left( {J_{v} + \kappa - J_{UF}} \right)}}{V_{p}^{2}} \right)}\frac{\partial V_{p}}{\partial\alpha}} + \frac{1}{V_{p}}}} \\{{{\left( {\frac{\partial J_{s}}{\partial c_{p}} - {c_{p}\frac{\partial J_{v}}{\partial c_{p}}} - \left( {J_{v} + \kappa - J_{UF}} \right)} \right)\frac{\partial c_{p}}{\partial\alpha}} + \frac{1}{V_{p}}}} \\{{{{\left( {\frac{\partial J_{s}}{\partial c_{i}} - {c_{p}\frac{\partial J_{v}}{\partial c_{i}}}} \right)\frac{\partial c_{i}}{\partial\alpha}} + \frac{c_{p}}{V_{p}}},}}\end{matrix}$ $\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial V_{i}}{\partial J_{UF}} \right)} = {\frac{\partial}{\partial J_{UF}}\left( \frac{{dV}_{i}}{dt} \right)}} \\{= {{\frac{\partial}{\partial V_{p}}\left( {{- J_{v}} - \kappa} \right)\frac{\partial V_{p}}{\partial J_{UF}}} + {\frac{\partial}{\partial c_{p}}\left( {{- J_{v}} - \kappa} \right)\frac{\partial c_{p}}{\partial J_{UF}}} +}} \\{{{\frac{\partial}{\partial V_{i}}\left( {{- J_{v}} - \kappa} \right)\frac{\partial V_{i}}{\partial J_{UF}}} + {\frac{\partial}{\partial c_{i}}\left( {{- J_{v}} - \kappa} \right)\frac{\partial c_{i}}{\partial J_{UF}}} +}} \\{{\frac{\partial}{\partial J_{UF}}\left( {{- J_{v}} - \kappa} \right)}} \\{{= {{{- \frac{\partial J_{v}}{\partial c_{p}}}\frac{\partial c_{p}}{\partial J_{UF}}} - {\frac{\partial J_{v}}{\partial c_{i}}\frac{\partial c_{i}}{\partial J_{UF}}}}},}\end{matrix}$ $\begin{matrix}{{\frac{d}{dt}\left( \frac{\partial c_{i}}{\partial J_{UF}} \right)} = {\frac{\partial}{\partial J_{UF}}\left( \frac{{{dc}_{i}}_{i}}{dt} \right)}} \\{= {{\frac{\partial}{\partial V_{p}}\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)\frac{\partial V_{p}}{\partial J_{UF}}} + \frac{\partial}{\partial J_{UF}}}} \\{{{\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)\frac{\partial c_{p}}{\partial J_{UF}}} + \frac{\partial}{\partial V_{i}}}} \\{{{\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)\frac{\partial V_{i}}{\partial J_{UF}}} + {\frac{\partial}{\partial c_{i}}\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)}}} \\{{\frac{\partial c_{i}}{\partial J_{UF}} + {\frac{\partial}{\partial J_{UF}}\left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}} \right)}}} \\{= {{\frac{1}{V_{1}}\left( {{- \frac{\partial J_{s}}{\partial c_{p}}} + {c_{i}\frac{\partial J_{v}}{\partial c_{p}}}} \right)\frac{\partial c_{p}}{\partial J_{UF}}} - \left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}^{2}} \right)}} \\{{\frac{\partial V_{i}}{\partial J_{UF}} + {\frac{1}{V_{i}}\left( {{- \frac{\partial J_{s}}{\partial c_{i}}} + {c_{i}\frac{\partial J_{v}}{\partial c_{i}}} + \left( {J_{v} + \kappa} \right)} \right)}}} \\{{\frac{\partial c_{i}}{\partial J_{UF}} + {\frac{1}{V_{i}}\left( {{- \frac{\partial J_{s}}{\partial J_{UF}}} + {c_{i}\frac{\partial J_{v}}{\partial J_{UF}}}} \right)}}} \\{= {{\frac{1}{V_{i}}\left( {{- \frac{\partial J_{s}}{\partial c_{p}}} + {c_{i}\frac{\partial J_{v}}{\partial c_{p}}}} \right)\frac{\partial c_{p}}{\partial J_{UF}}} - \left( \frac{{- J_{s}} + {c_{i}\left( {J_{v} + \kappa} \right)}}{V_{i}^{2}} \right)}} \\{{\frac{\partial V_{i}}{\partial J_{UF}} + {\frac{1}{V_{i}}\left( {{- \frac{\partial J_{s}}{\partial c_{i}}} + {c_{i}\frac{\partial J_{v}}{\partial c_{i}}} + \left( {J_{v} + \kappa} \right)} \right){\frac{\partial c_{i}}{\partial J_{UF}}.}}}}\end{matrix}$

1. A system, comprising: a dialysis machine configured to providehemodialysis treatment to a patient; a monitoring device configured toobtain hematocrit measurements corresponding to blood of the patientduring the hemodialysis treatment; and a processing system configuredto: receive the hematocrit measurements; determine an initial set ofparameter values to be used with a two-compartment vascular refillmodel; estimate values of one or more parameters for the patient basedon the hematocrit measurements, the initial set of parameter values, andthe two-compartment vascular refill model; and output the estimatedvalues for the patient.
 2. The system according to claim 1, wherein theone or more parameters for the patient include: a filtration coefficient(L_(p)); a hydrostatic capillary pressure (P_(c)); a hydrostaticinterstitial pressure (P_(i)); a systemic capillary reflectioncoefficient (σ); a constant protein concentration (α); and/or a constantlymph flow rate (κ).
 3. The system according to claim 1, whereindetermining the initial set of parameters comprises: determining whetheror not previous parameter estimates exist for the one or moreparameters; and based on the determination of whether or not previousparameter estimates exist for the one or more parameters, using theprevious parameter estimates or using predetermined parameter values asthe initial set of parameter values.
 4. The system according to claim 1,wherein estimating the values of the one or more parameters for thepatient based on the hematocrit measurements, the initial set ofparameters, and the two-compartment vascular refill model furthercomprises: performing a parameter identification; solving modelequations using the initial set of parameter values to determine whetherthe two-compartment vascular refill model fits the hematocritmeasurements; and in response to determining that the two-compartmentvascular refill model does not fit the hematocrit measurements,modifying the initial set of parameter values.
 5. The system accordingto claim 1, wherein estimating the values of the one or more parametersfor the patient based on the hematocrit measurements, the initial set ofparameters, and the two-compartment vascular refill model furthercomprises: performing a parameter identification; solving modelequations using the determined initial set of parameter values todetermine whether the two-compartment vascular refill model fits thehematocrit measurements; in response to determining that thetwo-compartment vascular refill model fits the hematocrit measurements,determining whether the initial set of parameter values are within apredetermined range; and in response to determining that the initial setof parameter values are not within the predetermined range, modifyingthe initial set of parameter values.
 6. The system according to claim 1,wherein estimating the values of the one or more parameters for thepatient based on the hematocrit measurements, the initial set ofparameters, and the two-compartment vascular refill model furthercomprises: performing a parameter identification; solving modelequations using the initial set of parameter values to determine whetherthe two-compartment vascular refill model fits the hematocritmeasurements; and in response to determining that the model fits thehematocrit measurements, determining whether the initial set ofparameter values are within a predetermined range; and in response todetermining that the initial set of parameter values are within thepredetermined range, using the initial set of parameter values as theestimated values for output.
 7. The system according to claim 1, whereinthe processing system is further configured to: output and store trenddata for the one or more parameters based on the estimated values; causemodifications to current and/or future treatments for the patient;and/or generate notifications and/or alerts based on the estimatedvalues.
 8. The system according to claim 1, wherein the two-compartmentvascular refill model includes model equations defining: change inplasma volume as a function of time; change in interstitial volume as afunction of time; change in plasma protein concentration as a functionof time; and change in interstitial protein concentration as a functionof time.
 9. The system according to claim 8, wherein the two-compartmentvascular refill model further includes model equations defining: anamount of fluid crossing a capillary membrane as a function of time; andnet flux of proteins between plasma and interstitium as a function oftime.
 10. A method, comprising: receiving, by a processing system,hematocrit measurements corresponding to blood of a patient duringhemodialysis treatment for the patient; determining, by the processingsystem, an initial set of parameter values to be used with atwo-compartment vascular refill model; estimating, by the processingsystem, values of one or more parameters for the patient based on thehematocrit measurements, the initial set of parameter values, and thetwo-compartment vascular refill model; and outputting, by the processingsystem, the estimated values for the patient.
 11. The method accordingto claim 10, wherein the one or more parameters for the patient include:a filtration coefficient (L_(p)); a hydrostatic capillary pressure(P_(c)); a hydrostatic interstitial pressure (P_(i)); a systemiccapillary reflection coefficient (σ); a constant protein concentration(α); and/or a constant lymph flow rate (κ).
 12. The method according toclaim 10, wherein determining the initial set of parameters comprises:determining whether or not previous parameter estimates exist for theone or more parameters; and based on the determination of whether or notprevious parameter estimates exist for the one or more parameters, usingthe previous parameter estimates or using predetermined parameter valuesas the initial set of parameter values.
 13. The method according toclaim 10, wherein estimating the values of the one or more parametersfor the patient based on the hematocrit measurements, the initial set ofparameters, and the two-compartment vascular refill model furthercomprises: performing a parameter identification; solving modelequations using the initial set of parameter values to determine whetherthe two-compartment vascular refill model fits the hematocritmeasurements; and in response to determining that the two-compartmentvascular refill model does not fit the hematocrit measurements,modifying the initial set of parameter values.
 14. The method accordingto claim 10, wherein estimating the values of the one or more parametersfor the patient based on the hematocrit measurements, the initial set ofparameters, and the two-compartment vascular refill model furthercomprises: performing a parameter identification; solving modelequations using the determined initial set of parameter values todetermine whether the two-compartment vascular refill model fits thehematocrit measurements; in response to determining that thetwo-compartment vascular refill model fits the hematocrit measurements,determining whether the initial set of parameter values are within apredetermined range; and in response to determining that the initial setof parameter values are not within the predetermined range, modifyingthe initial set of parameter values.
 15. The method according to claim10, wherein estimating the values of the one or more parameters for thepatient based on the hematocrit measurements, the initial set ofparameters, and the two-compartment vascular refill model furthercomprises: performing a parameter identification; solving modelequations using the initial set of parameter values to determine whetherthe two-compartment vascular refill model fits the hematocritmeasurements; and in response to determining that the model fits thehematocrit measurements, determining whether the initial set ofparameter values are within a predetermined range; and in response todetermining that the initial set of parameter values are within thepredetermined range, using the initial set of parameter values as theestimated values for output.
 16. The method according to claim 10,further comprising: outputting and storing trend data for the one ormore parameters based on the estimated values; modifying current and/orfuture treatments for the patient; and/or generating notificationsand/or alerts based on the estimated values.
 17. The method according toclaim 10, wherein the two-compartment vascular refill model includesmodel equations defining: change in plasma volume as a function of time;change in interstitial volume as a function of time; change in plasmaprotein concentration as a function of time; and change in interstitialprotein concentration as a function of time.
 18. The method according toclaim 10, wherein the two-compartment vascular refill model furtherincludes model equations defining: an amount of fluid crossing acapillary membrane as a function of time; and net flux of proteinsbetween plasma and interstitium as a function of time.
 19. Anon-transitory processor-readable medium having processor-executableinstructions stored thereon, wherein the processor-executableinstructions, when executed, facilitate: receiving hematocritmeasurements corresponding to blood of a patient during hemodialysistreatment for the patient; determining an initial set of parametervalues to be used with a two-compartment vascular refill model;estimating values of one or more parameters for the patient based on thehematocrit measurements, the initial set of parameter values, and thetwo-compartment vascular refill model; and outputting the estimatedvalues for the patient.
 20. The non-transitory processor-readable mediumaccording to claim 19, wherein estimating the values of the one or moreparameters for the patient based on the hematocrit measurements, theinitial set of parameters, and the two-compartment vascular refill modelfurther comprises: performing a parameter identification; solving modelequations using the initial set of parameter values to determine whetherthe two-compartment vascular refill model fits the hematocritmeasurements; and in response to determining that the two-compartmentvascular refill model does not fit the hematocrit measurements,modifying the initial set of parameter values.
 21. The non-transitoryprocessor-readable medium according to claim 19, wherein thetwo-compartment vascular refill model includes model equations defining:change in plasma volume as a function of time; change in interstitialvolume as a function of time; change in plasma protein concentration asa function of time; change in interstitial protein concentration as afunction of time; an amount of fluid crossing a capillary membrane as afunction of time; and net flux of proteins between plasma andinterstitium as a function of time.